Special functions of several complex variables
Jacobi's theta functionθ 1 q =e i πτ e 0.1i π θ 1 ( z , q ) = 2 q 1 4 ∑ n = 0 ∞ ( − 1 ) n q n ( n + 1 ) sin ( 2 n + 1 ) z = ∑ n = − ∞ ∞ ( − 1 ) n − 1 2 q ( n + 1 2 ) 2 e ( 2 n + 1 ) i z . {\displaystyle {\begin{aligned}\theta _{1}(z,q)&=2q^{\frac {1}{4}}\sum _{n=0}^{\infty }(-1)^{n}q^{n(n+1)}\sin(2n+1)z\\&=\sum _{n=-\infty }^{\infty }(-1)^{n-{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}e^{(2n+1)iz}.\end{aligned}}} Inmathematics ,theta functions arespecial functions ofseveral complex variables . They show up in many topics, includingAbelian varieties ,moduli spaces ,quadratic forms , andsolitons . Theta functions are parametrized by points in atube domain inside a complexLagrangian Grassmannian ,[ 1] Siegel upper half space .
The most common form of theta function is that occurring in the theory ofelliptic functions . With respect to one of the complex variables (conventionally calledz ), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it aquasiperiodic function . In the abstract theory this quasiperiodicity comes from thecohomology class of aline bundle on a complex torus , a condition ofdescent .
One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".[ 2]
Throughout this article,( e π i τ ) α {\displaystyle (e^{\pi i\tau })^{\alpha }} e α π i τ {\displaystyle e^{\alpha \pi i\tau }} branch ).[ note 1]
Jacobi theta function [ edit ] There are several closely related functions called Jacobi theta functions, andmany different and incompatible systems of notation for them. OneJacobi theta function (named afterCarl Gustav Jacob Jacobi ) is a function defined for two complex variablesz andτ , wherez can be anycomplex number andτ is thehalf-period ratio , confined to theupper half-plane , which means it has a positive imaginary part. It is given by the formula
ϑ ( z ; τ ) = ∑ n = − ∞ ∞ exp ( π i n 2 τ + 2 π i n z ) = 1 + 2 ∑ n = 1 ∞ q n 2 cos ( 2 π n z ) = ∑ n = − ∞ ∞ q n 2 η n {\displaystyle {\begin{aligned}\vartheta (z;\tau )&=\sum _{n=-\infty }^{\infty }\exp \left(\pi in^{2}\tau +2\pi inz\right)\\&=1+2\sum _{n=1}^{\infty }q^{n^{2}}\cos(2\pi nz)\\&=\sum _{n=-\infty }^{\infty }q^{n^{2}}\eta ^{n}\end{aligned}}} whereq = exp(πiτ )nome andη = exp(2πiz )Jacobi form . The restriction ensures that it is an absolutely convergent series. At fixedτ , this is aFourier series for a 1-periodicentire function ofz . Accordingly, the theta function is 1-periodic inz :
ϑ ( z + 1 ; τ ) = ϑ ( z ; τ ) . {\displaystyle \vartheta (z+1;\tau )=\vartheta (z;\tau ).} Bycompleting the square , it is alsoτ -quasiperiodic inz , with
ϑ ( z + τ ; τ ) = exp ( − π i ( τ + 2 z ) ) ϑ ( z ; τ ) . {\displaystyle \vartheta (z+\tau ;\tau )=\exp {\bigl (}-\pi i(\tau +2z){\bigr )}\vartheta (z;\tau ).} Thus, in general,
ϑ ( z + a + b τ ; τ ) = exp ( − π i b 2 τ − 2 π i b z ) ϑ ( z ; τ ) {\displaystyle \vartheta (z+a+b\tau ;\tau )=\exp \left(-\pi ib^{2}\tau -2\pi ibz\right)\vartheta (z;\tau )} for any integersa andb .
For any fixedτ {\displaystyle \tau } Liouville's theorem , it cannot be doubly periodic in1 , τ {\displaystyle 1,\tau } 1 {\displaystyle 1} τ {\displaystyle \tau } | ϑ ( z + a + b τ ; τ ) ϑ ( z ; τ ) | = exp ( π ( b 2 ℑ ( τ ) + 2 b ℑ ( z ) ) ) {\displaystyle \left|{\frac {\vartheta (z+a+b\tau ;\tau )}{\vartheta (z;\tau )}}\right|=\exp \left(\pi (b^{2}\Im (\tau )+2b\Im (z))\right)} ℑ ( τ ) > 0 {\displaystyle \Im (\tau )>0} ϑ ( z , τ ) {\displaystyle \vartheta (z,\tau )}
It is in fact the most general entire function with 2 quasi-periods, in the following sense:[ 3]
Theta functionθ 1 q =e iπτ q changes withτ . Theta functionθ 1 q =e iπτ q changes withτ . Auxiliary functions [ edit ] The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript:
ϑ 00 ( z ; τ ) = ϑ ( z ; τ ) {\displaystyle \vartheta _{00}(z;\tau )=\vartheta (z;\tau )} The auxiliary (or half-period) functions are defined by
ϑ 01 ( z ; τ ) = ϑ ( z + 1 2 ; τ ) ϑ 10 ( z ; τ ) = exp ( 1 4 π i τ + π i z ) ϑ ( z + 1 2 τ ; τ ) ϑ 11 ( z ; τ ) = exp ( 1 4 π i τ + π i ( z + 1 2 ) ) ϑ ( z + 1 2 τ + 1 2 ; τ ) . {\displaystyle {\begin{aligned}\vartheta _{01}(z;\tau )&=\vartheta \left(z+{\tfrac {1}{2}};\tau \right)\\[3pt]\vartheta _{10}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi iz\right)\vartheta \left(z+{\tfrac {1}{2}}\tau ;\tau \right)\\[3pt]\vartheta _{11}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi i\left(z+{\tfrac {1}{2}}\right)\right)\vartheta \left(z+{\tfrac {1}{2}}\tau +{\tfrac {1}{2}};\tau \right).\end{aligned}}} This notation followsRiemann andMumford ;Jacobi 's original formulation was in terms of thenome q =e iπτ τ . In Jacobi's notation theθ -functions are written:
θ 1 ( z ; q ) = θ 1 ( π z , q ) = − ϑ 11 ( z ; τ ) θ 2 ( z ; q ) = θ 2 ( π z , q ) = ϑ 10 ( z ; τ ) θ 3 ( z ; q ) = θ 3 ( π z , q ) = ϑ 00 ( z ; τ ) θ 4 ( z ; q ) = θ 4 ( π z , q ) = ϑ 01 ( z ; τ ) {\displaystyle {\begin{aligned}\theta _{1}(z;q)&=\theta _{1}(\pi z,q)=-\vartheta _{11}(z;\tau )\\\theta _{2}(z;q)&=\theta _{2}(\pi z,q)=\vartheta _{10}(z;\tau )\\\theta _{3}(z;q)&=\theta _{3}(\pi z,q)=\vartheta _{00}(z;\tau )\\\theta _{4}(z;q)&=\theta _{4}(\pi z,q)=\vartheta _{01}(z;\tau )\end{aligned}}} Jacobi theta 1 Jacobi theta 2 Jacobi theta 3 Jacobi theta 4 The above definitions of the Jacobi theta functions are by no means unique. SeeJacobi theta functions (notational variations) for further discussion.
If we setz = 0τ only, defined on the upper half-plane. These functions are calledTheta Nullwert functions, based on the German term forzero value because of the annullation of the left entry in the theta function expression. Alternatively, we obtain four functions ofq only, defined on the unit disk| q | < 1 {\displaystyle |q|<1} theta constants :[ note 2]
ϑ 11 ( 0 ; τ ) = − θ 1 ( q ) = − ∑ n = − ∞ ∞ ( − 1 ) n − 1 / 2 q ( n + 1 / 2 ) 2 ϑ 10 ( 0 ; τ ) = θ 2 ( q ) = ∑ n = − ∞ ∞ q ( n + 1 / 2 ) 2 ϑ 00 ( 0 ; τ ) = θ 3 ( q ) = ∑ n = − ∞ ∞ q n 2 ϑ 01 ( 0 ; τ ) = θ 4 ( q ) = ∑ n = − ∞ ∞ ( − 1 ) n q n 2 {\displaystyle {\begin{aligned}\vartheta _{11}(0;\tau )&=-\theta _{1}(q)=-\sum _{n=-\infty }^{\infty }(-1)^{n-1/2}q^{(n+1/2)^{2}}\\\vartheta _{10}(0;\tau )&=\theta _{2}(q)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}\\\vartheta _{00}(0;\tau )&=\theta _{3}(q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\\\vartheta _{01}(0;\tau )&=\theta _{4}(q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}\end{aligned}}} with thenome q =e iπτ θ 1 ( q ) = 0 {\displaystyle \theta _{1}(q)=0} modular forms , and to parametrize certain curves; in particular, theJacobi identity is
θ 2 ( q ) 4 + θ 4 ( q ) 4 = θ 3 ( q ) 4 {\displaystyle \theta _{2}(q)^{4}+\theta _{4}(q)^{4}=\theta _{3}(q)^{4}} or equivalently,
ϑ 01 ( 0 ; τ ) 4 + ϑ 10 ( 0 ; τ ) 4 = ϑ 00 ( 0 ; τ ) 4 {\displaystyle \vartheta _{01}(0;\tau )^{4}+\vartheta _{10}(0;\tau )^{4}=\vartheta _{00}(0;\tau )^{4}} which is theFermat curve of degree four.
Jacobi's identities describe how theta functions transform under themodular group , which is generated byτ ↦τ + 1τ ↦ −1 / τ τ in the exponent has the same effect as adding1 / 2 toz (n ≡n 2 mod 2
α = ( − i τ ) 1 2 exp ( π τ i z 2 ) . {\displaystyle \alpha =(-i\tau )^{\frac {1}{2}}\exp \left({\frac {\pi }{\tau }}iz^{2}\right).} Then
ϑ 00 ( z τ ; − 1 τ ) = α ϑ 00 ( z ; τ ) ϑ 01 ( z τ ; − 1 τ ) = α ϑ 10 ( z ; τ ) ϑ 10 ( z τ ; − 1 τ ) = α ϑ 01 ( z ; τ ) ϑ 11 ( z τ ; − 1 τ ) = − i α ϑ 11 ( z ; τ ) . {\displaystyle {\begin{aligned}\vartheta _{00}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{00}(z;\tau )\quad &\vartheta _{01}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{10}(z;\tau )\\[3pt]\vartheta _{10}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{01}(z;\tau )\quad &\vartheta _{11}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=-i\alpha \,\vartheta _{11}(z;\tau ).\end{aligned}}} Theta functions in terms of the nome [ edit ] Instead of expressing the Theta functions in terms ofz andτ , we may express them in terms of argumentsw and thenome q , wherew =e πiz q =e πiτ
ϑ 00 ( w , q ) = ∑ n = − ∞ ∞ ( w 2 ) n q n 2 ϑ 01 ( w , q ) = ∑ n = − ∞ ∞ ( − 1 ) n ( w 2 ) n q n 2 ϑ 10 ( w , q ) = ∑ n = − ∞ ∞ ( w 2 ) n + 1 2 q ( n + 1 2 ) 2 ϑ 11 ( w , q ) = i ∑ n = − ∞ ∞ ( − 1 ) n ( w 2 ) n + 1 2 q ( n + 1 2 ) 2 . {\displaystyle {\begin{aligned}\vartheta _{00}(w,q)&=\sum _{n=-\infty }^{\infty }\left(w^{2}\right)^{n}q^{n^{2}}\quad &\vartheta _{01}(w,q)&=\sum _{n=-\infty }^{\infty }(-1)^{n}\left(w^{2}\right)^{n}q^{n^{2}}\\[3pt]\vartheta _{10}(w,q)&=\sum _{n=-\infty }^{\infty }\left(w^{2}\right)^{n+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}\quad &\vartheta _{11}(w,q)&=i\sum _{n=-\infty }^{\infty }(-1)^{n}\left(w^{2}\right)^{n+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}.\end{aligned}}} We see that the theta functions can also be defined in terms ofw andq , without a direct reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over otherfields where the exponential function might not be everywhere defined, such as fields ofp -adic numbers
Product representations [ edit ] TheJacobi triple product (a special case of theMacdonald identities ) tells us that for complex numbersw andq with|q andw ≠ 0
∏ m = 1 ∞ ( 1 − q 2 m ) ( 1 + w 2 q 2 m − 1 ) ( 1 + w − 2 q 2 m − 1 ) = ∑ n = − ∞ ∞ w 2 n q n 2 . {\displaystyle \prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+w^{2}q^{2m-1}\right)\left(1+w^{-2}q^{2m-1}\right)=\sum _{n=-\infty }^{\infty }w^{2n}q^{n^{2}}.} It can be proven by elementary means, as for instance in Hardy and Wright'sAn Introduction to the Theory of Numbers
If we express the theta function in terms of the nomeq =e πiτ q =e 2πiτ w =e πiz
ϑ ( z ; τ ) = ∑ n = − ∞ ∞ exp ( π i τ n 2 ) exp ( 2 π i z n ) = ∑ n = − ∞ ∞ w 2 n q n 2 . {\displaystyle \vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi i\tau n^{2})\exp(2\pi izn)=\sum _{n=-\infty }^{\infty }w^{2n}q^{n^{2}}.} We therefore obtain a product formula for the theta function in the form
ϑ ( z ; τ ) = ∏ m = 1 ∞ ( 1 − exp ( 2 m π i τ ) ) ( 1 + exp ( ( 2 m − 1 ) π i τ + 2 π i z ) ) ( 1 + exp ( ( 2 m − 1 ) π i τ − 2 π i z ) ) . {\displaystyle \vartheta (z;\tau )=\prod _{m=1}^{\infty }{\big (}1-\exp(2m\pi i\tau ){\big )}{\Big (}1+\exp {\big (}(2m-1)\pi i\tau +2\pi iz{\big )}{\Big )}{\Big (}1+\exp {\big (}(2m-1)\pi i\tau -2\pi iz{\big )}{\Big )}.} In terms ofw andq :
ϑ ( z ; τ ) = ∏ m = 1 ∞ ( 1 − q 2 m ) ( 1 + q 2 m − 1 w 2 ) ( 1 + q 2 m − 1 w 2 ) = ( q 2 ; q 2 ) ∞ ( − w 2 q ; q 2 ) ∞ ( − q w 2 ; q 2 ) ∞ = ( q 2 ; q 2 ) ∞ θ ( − w 2 q ; q 2 ) {\displaystyle {\begin{aligned}\vartheta (z;\tau )&=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+q^{2m-1}w^{2}\right)\left(1+{\frac {q^{2m-1}}{w^{2}}}\right)\\&=\left(q^{2};q^{2}\right)_{\infty }\,\left(-w^{2}q;q^{2}\right)_{\infty }\,\left(-{\frac {q}{w^{2}}};q^{2}\right)_{\infty }\\&=\left(q^{2};q^{2}\right)_{\infty }\,\theta \left(-w^{2}q;q^{2}\right)\end{aligned}}} where( ; )∞ is theq -Pochhammer symbolθ ( ; )q -theta function
∏ m = 1 ∞ ( 1 − q 2 m ) ( 1 + ( w 2 + w − 2 ) q 2 m − 1 + q 4 m − 2 ) , {\displaystyle \prod _{m=1}^{\infty }\left(1-q^{2m}\right){\Big (}1+\left(w^{2}+w^{-2}\right)q^{2m-1}+q^{4m-2}{\Big )},} which we may also write as
ϑ ( z ∣ q ) = ∏ m = 1 ∞ ( 1 − q 2 m ) ( 1 + 2 cos ( 2 π z ) q 2 m − 1 + q 4 m − 2 ) . {\displaystyle \vartheta (z\mid q)=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q^{2m-1}+q^{4m-2}\right).} This form is valid in general but clearly is of particular interest whenz is real. Similar product formulas for the auxiliary theta functions are
ϑ 01 ( z ∣ q ) = ∏ m = 1 ∞ ( 1 − q 2 m ) ( 1 − 2 cos ( 2 π z ) q 2 m − 1 + q 4 m − 2 ) , ϑ 10 ( z ∣ q ) = 2 q 1 4 cos ( π z ) ∏ m = 1 ∞ ( 1 − q 2 m ) ( 1 + 2 cos ( 2 π z ) q 2 m + q 4 m ) , ϑ 11 ( z ∣ q ) = − 2 q 1 4 sin ( π z ) ∏ m = 1 ∞ ( 1 − q 2 m ) ( 1 − 2 cos ( 2 π z ) q 2 m + q 4 m ) . {\displaystyle {\begin{aligned}\vartheta _{01}(z\mid q)&=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m-1}+q^{4m-2}\right),\\[3pt]\vartheta _{10}(z\mid q)&=2q^{\frac {1}{4}}\cos(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q^{2m}+q^{4m}\right),\\[3pt]\vartheta _{11}(z\mid q)&=-2q^{\frac {1}{4}}\sin(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m}+q^{4m}\right).\end{aligned}}} In particular,lim q → 0 ϑ 10 ( z ∣ q ) 2 q 1 4 = cos ( π z ) , lim q → 0 − ϑ 11 ( z ∣ q ) 2 q − 1 4 = sin ( π z ) {\displaystyle \lim _{q\to 0}{\frac {\vartheta _{10}(z\mid q)}{2q^{\frac {1}{4}}}}=\cos(\pi z),\quad \lim _{q\to 0}{\frac {-\vartheta _{11}(z\mid q)}{2q^{-{\frac {1}{4}}}}}=\sin(\pi z)} sin , cos {\displaystyle \sin ,\cos }
Integral representations [ edit ] The Jacobi theta functions have the following integral representations:
ϑ 00 ( z ; τ ) = − i ∫ i − ∞ i + ∞ e i π τ u 2 cos ( 2 π u z + π u ) sin ( π u ) d u ; ϑ 01 ( z ; τ ) = − i ∫ i − ∞ i + ∞ e i π τ u 2 cos ( 2 π u z ) sin ( π u ) d u ; ϑ 10 ( z ; τ ) = − i e i π z + 1 4 i π τ ∫ i − ∞ i + ∞ e i π τ u 2 cos ( 2 π u z + π u + π τ u ) sin ( π u ) d u ; ϑ 11 ( z ; τ ) = e i π z + 1 4 i π τ ∫ i − ∞ i + ∞ e i π τ u 2 cos ( 2 π u z + π τ u ) sin ( π u ) d u . {\displaystyle {\begin{aligned}\vartheta _{00}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz+\pi u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{01}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{10}(z;\tau )&=-ie^{i\pi z+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz+\pi u+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{11}(z;\tau )&=e^{i\pi z+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u.\end{aligned}}} The Theta Nullwert functionθ 3 ( q ) {\displaystyle \theta _{3}(q)}
θ 3 ( q ) = 1 + 4 q ln ( 1 / q ) π ∫ 0 ∞ exp [ − ln ( 1 / q ) x 2 ] { 1 − q 2 cos [ 2 ln ( 1 / q ) x ] } 1 − 2 q 2 cos [ 2 ln ( 1 / q ) x ] + q 4 d x {\displaystyle \theta _{3}(q)=1+{\frac {4q{\sqrt {\ln(1/q)}}}{\sqrt {\pi }}}\int _{0}^{\infty }{\frac {\exp[-\ln(1/q)\,x^{2}]\{1-q^{2}\cos[2\ln(1/q)\,x]\}}{1-2q^{2}\cos[2\ln(1/q)\,x]+q^{4}}}\,\mathrm {d} x} This formula was discussed in the essaySquare series generating function transformations by the mathematician Maxie Schmidt from Georgia in Atlanta.
Based on this formula following three eminent examples are given:
[ 2 π K ( 1 2 2 ) ] 1 / 2 = θ 3 [ exp ( − π ) ] = 1 + 4 exp ( − π ) ∫ 0 ∞ exp ( − π x 2 ) [ 1 − exp ( − 2 π ) cos ( 2 π x ) ] 1 − 2 exp ( − 2 π ) cos ( 2 π x ) + exp ( − 4 π ) d x {\displaystyle {\biggl [}{\frac {2}{\pi }}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}{\biggr ]}^{1/2}=\theta _{3}{\bigl [}\exp(-\pi ){\bigr ]}=1+4\exp(-\pi )\int _{0}^{\infty }{\frac {\exp(-\pi x^{2})[1-\exp(-2\pi )\cos(2\pi x)]}{1-2\exp(-2\pi )\cos(2\pi x)+\exp(-4\pi )}}\,\mathrm {d} x} [ 2 π K ( 2 − 1 ) ] 1 / 2 = θ 3 [ exp ( − 2 π ) ] = 1 + 4 2 4 exp ( − 2 π ) ∫ 0 ∞ exp ( − 2 π x 2 ) [ 1 − exp ( − 2 2 π ) cos ( 2 2 π x ) ] 1 − 2 exp ( − 2 2 π ) cos ( 2 2 π x ) + exp ( − 4 2 π ) d x {\displaystyle {\biggl [}{\frac {2}{\pi }}K({\sqrt {2}}-1){\biggr ]}^{1/2}=\theta _{3}{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}=1+4\,{\sqrt[{4}]{2}}\exp(-{\sqrt {2}}\,\pi )\int _{0}^{\infty }{\frac {\exp(-{\sqrt {2}}\,\pi x^{2})[1-\exp(-2{\sqrt {2}}\,\pi )\cos(2{\sqrt {2}}\,\pi x)]}{1-2\exp(-2{\sqrt {2}}\,\pi )\cos(2{\sqrt {2}}\,\pi x)+\exp(-4{\sqrt {2}}\,\pi )}}\,\mathrm {d} x} { 2 π K [ sin ( π 12 ) ] } 1 / 2 = θ 3 [ exp ( − 3 π ) ] = 1 + 4 3 4 exp ( − 3 π ) ∫ 0 ∞ exp ( − 3 π x 2 ) [ 1 − exp ( − 2 3 π ) cos ( 2 3 π x ) ] 1 − 2 exp ( − 2 3 π ) cos ( 2 3 π x ) + exp ( − 4 3 π ) d x {\displaystyle {\biggl \{}{\frac {2}{\pi }}K{\bigl [}\sin {\bigl (}{\frac {\pi }{12}}{\bigr )}{\bigr ]}{\biggr \}}^{1/2}=\theta _{3}{\bigl [}\exp(-{\sqrt {3}}\,\pi ){\bigr ]}=1+4\,{\sqrt[{4}]{3}}\exp(-{\sqrt {3}}\,\pi )\int _{0}^{\infty }{\frac {\exp(-{\sqrt {3}}\,\pi x^{2})[1-\exp(-2{\sqrt {3}}\,\pi )\cos(2{\sqrt {3}}\,\pi x)]}{1-2\exp(-2{\sqrt {3}}\,\pi )\cos(2{\sqrt {3}}\,\pi x)+\exp(-4{\sqrt {3}}\,\pi )}}\,\mathrm {d} x} Furthermore, the theta examplesθ 3 ( 1 2 ) {\displaystyle \theta _{3}({\tfrac {1}{2}})} θ 3 ( 1 3 ) {\displaystyle \theta _{3}({\tfrac {1}{3}})}
θ 3 ( 1 2 ) = 1 + 2 ∑ n = 1 ∞ 1 2 n 2 = 1 + 2 π − 1 / 2 ln ( 2 ) ∫ 0 ∞ exp [ − ln ( 2 ) x 2 ] { 16 − 4 cos [ 2 ln ( 2 ) x ] } 17 − 8 cos [ 2 ln ( 2 ) x ] d x {\displaystyle \theta _{3}\left({\frac {1}{2}}\right)=1+2\sum _{n=1}^{\infty }{\frac {1}{2^{n^{2}}}}=1+2\pi ^{-1/2}{\sqrt {\ln(2)}}\int _{0}^{\infty }{\frac {\exp[-\ln(2)\,x^{2}]\{16-4\cos[2\ln(2)\,x]\}}{17-8\cos[2\ln(2)\,x]}}\,\mathrm {d} x} θ 3 ( 1 2 ) = 2.128936827211877158669 … {\displaystyle \theta _{3}\left({\frac {1}{2}}\right)=2.128936827211877158669\ldots } θ 3 ( 1 3 ) = 1 + 2 ∑ n = 1 ∞ 1 3 n 2 = 1 + 4 3 π − 1 / 2 ln ( 3 ) ∫ 0 ∞ exp [ − ln ( 3 ) x 2 ] { 81 − 9 cos [ 2 ln ( 3 ) x ] } 82 − 18 cos [ 2 ln ( 3 ) x ] d x {\displaystyle \theta _{3}\left({\frac {1}{3}}\right)=1+2\sum _{n=1}^{\infty }{\frac {1}{3^{n^{2}}}}=1+{\frac {4}{3}}\pi ^{-1/2}{\sqrt {\ln(3)}}\int _{0}^{\infty }{\frac {\exp[-\ln(3)\,x^{2}]\{81-9\cos[2\ln(3)\,x]\}}{82-18\cos[2\ln(3)\,x]}}\,\mathrm {d} x} θ 3 ( 1 3 ) = 1.691459681681715341348 … {\displaystyle \theta _{3}\left({\frac {1}{3}}\right)=1.691459681681715341348\ldots } Some interesting relations [ edit ] If| q | < 1 {\displaystyle |q|<1} a > 0 {\displaystyle a>0}
θ 3 ( a , b ; q ) = ∑ n = − ∞ ∞ q a n 2 + b n {\displaystyle \theta _{3}(a,b;q)=\sum _{n=-\infty }^{\infty }q^{an^{2}+bn}} θ 4 ( a , b ; q ) = ∑ n = − ∞ ∞ ( − 1 ) n q a n 2 + b n {\displaystyle \theta _{4}(a,b;q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{an^{2}+bn}} have interesting arithmetical and modular properties. Whena , b , p {\displaystyle a,b,p} [ 4] [ 5]
log ( θ 3 ( p 2 , p 2 − a ; q ) θ 3 ( p 2 , p 2 − b ; q ) ) = − ∑ n = 1 ∞ q n ( ∑ d | n n / d ≡ ± a ( p ) ( − 1 ) d d − ∑ d | n n / d ≡ ± b ( p ) ( − 1 ) d d ) {\displaystyle \log \left({\frac {\theta _{3}\left({\frac {p}{2}},{\frac {p}{2}}-a;q\right)}{\theta _{3}\left({\frac {p}{2}},{\frac {p}{2}}-b;q\right)}}\right)=-\sum _{n=1}^{\infty }q^{n}\left(\sum _{\begin{array}{cc}d|n\\n/d\equiv \pm a(p)\end{array}}{\frac {(-1)^{d}}{d}}-\sum _{\begin{array}{cc}d|n\\n/d\equiv \pm b(p)\end{array}}{\frac {(-1)^{d}}{d}}\right)} log ( θ 4 ( p 2 , p 2 − a ; q ) θ 4 ( p 2 , p 2 − b ; q ) ) = − ∑ n = 1 ∞ q n ( ∑ d | n n / d ≡ ± a ( p ) 1 d − ∑ d | n n / d ≡ ± b ( p ) 1 d ) {\displaystyle \log \left({\frac {\theta _{4}\left({\frac {p}{2}},{\frac {p}{2}}-a;q\right)}{\theta _{4}\left({\frac {p}{2}},{\frac {p}{2}}-b;q\right)}}\right)=-\sum _{n=1}^{\infty }q^{n}\left(\sum _{\begin{array}{cc}d|n\\n/d\equiv \pm a(p)\end{array}}{\frac {1}{d}}-\sum _{\begin{array}{cc}d|n\\n/d\equiv \pm b(p)\end{array}}{\frac {1}{d}}\right)} Also ifq = e π i z {\displaystyle q=e^{\pi iz}} I m ( z ) > 0 {\displaystyle Im(z)>0}
ϑ + ( z ) = θ + ( a , p ; z ) = q p / 8 + a 2 / ( 2 p ) − a / 2 θ 3 ( p 2 , p 2 − a ; q ) {\displaystyle \vartheta _{+}(z)=\theta _{+}(a,p;z)=q^{p/8+a^{2}/(2p)-a/2}\theta _{3}\left({\frac {p}{2}},{\frac {p}{2}}-a;q\right)} and
ϑ − ( z ) = θ − ( a , p ; z ) = q p / 8 + a 2 / ( 2 p ) − a / 2 θ 4 ( p 2 , p 2 − a ; q ) {\displaystyle \vartheta _{-}(z)=\theta _{-}(a,p;z)=q^{p/8+a^{2}/(2p)-a/2}\theta _{4}\left({\frac {p}{2}},{\frac {p}{2}}-a;q\right)} are modular forms with weight1 / 2 {\displaystyle 1/2} Γ ( 2 p ) {\displaystyle \Gamma (2p)} a 1 , b 1 , c 1 , d 1 {\displaystyle a_{1},b_{1},c_{1},d_{1}} a 1 , d 1 ≡ 1 ( 2 p ) {\displaystyle a_{1},d_{1}\equiv 1(2p)} b 1 , c 1 ≡ 0 ( 2 p ) {\displaystyle b_{1},c_{1}\equiv 0(2p)} a 1 d 1 − b 1 c 1 = 1 {\displaystyle a_{1}d_{1}-b_{1}c_{1}=1} ϵ ± = ϵ ± ( a 1 , b 1 , c 1 , d 1 ) {\displaystyle \epsilon _{\pm }=\epsilon _{\pm }(a_{1},b_{1},c_{1},d_{1})} ( ϵ ± ) 24 = 1 {\displaystyle (\epsilon _{\pm })^{24}=1} z {\displaystyle z} I m ( z ) > 0 {\displaystyle Im(z)>0}
ϑ ± ( a 1 z + b 1 c 1 z + d 1 ) = ϵ ± c 1 z + d 1 ϑ ± ( z ) {\displaystyle \vartheta _{\pm }\left({\frac {a_{1}z+b_{1}}{c_{1}z+d_{1}}}\right)=\epsilon _{\pm }{\sqrt {c_{1}z+d_{1}}}\vartheta _{\pm }(z)} Proper credit for most of these results goes to Ramanujan. SeeRamanujan's lost notebook and a relevant reference atEuler function . The Ramanujan results quoted atEuler function plus a few elementary operations give the results below, so they are either in Ramanujan's lost notebook or follow immediately from it. See also Yi (2004).[ 6]
φ ( q ) = ϑ 00 ( 0 ; τ ) = θ 3 ( 0 ; q ) = ∑ n = − ∞ ∞ q n 2 {\displaystyle \quad \varphi (q)=\vartheta _{00}(0;\tau )=\theta _{3}(0;q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}} with the nomeq = e π i τ , {\displaystyle q=e^{\pi i\tau },} τ = n − 1 , {\displaystyle \tau =n{\sqrt {-1}},} Dedekind eta function η ( τ ) . {\displaystyle \eta (\tau ).} n = 1 , 2 , 3 , … {\displaystyle n=1,2,3,\dots }
φ ( e − π ) = π 4 Γ ( 3 4 ) = 2 η ( − 1 ) φ ( e − 2 π ) = π 4 Γ ( 3 4 ) 2 + 2 2 φ ( e − 3 π ) = π 4 Γ ( 3 4 ) 1 + 3 108 8 φ ( e − 4 π ) = π 4 Γ ( 3 4 ) 2 + 8 4 4 φ ( e − 5 π ) = π 4 Γ ( 3 4 ) 2 + 5 5 φ ( e − 6 π ) = π 4 Γ ( 3 4 ) 1 4 + 3 4 + 4 4 + 9 4 12 3 8 φ ( e − 7 π ) = π 4 Γ ( 3 4 ) 13 + 7 + 7 + 3 7 14 3 8 ⋅ 7 16 φ ( e − 8 π ) = π 4 Γ ( 3 4 ) 2 + 2 + 128 8 4 φ ( e − 9 π ) = π 4 Γ ( 3 4 ) 1 + 2 + 2 3 3 3 φ ( e − 10 π ) = π 4 Γ ( 3 4 ) 64 4 + 80 4 + 81 4 + 100 4 200 4 φ ( e − 11 π ) = π 4 Γ ( 3 4 ) 11 + 11 + ( 5 + 3 3 + 11 + 33 ) − 44 + 33 3 3 + ( − 5 + 3 3 − 11 + 33 ) 44 + 33 3 3 52180524 8 φ ( e − 12 π ) = π 4 Γ ( 3 4 ) 1 4 + 2 4 + 3 4 + 4 4 + 9 4 + 18 4 + 24 4 2 108 8 φ ( e − 13 π ) = π 4 Γ ( 3 4 ) 13 + 8 13 + ( 11 − 6 3 + 13 ) 143 + 78 3 3 + ( 11 + 6 3 + 13 ) 143 − 78 3 3 19773 4 φ ( e − 14 π ) = π 4 Γ ( 3 4 ) 13 + 7 + 7 + 3 7 + 10 + 2 7 + 28 8 4 + 7 28 7 16 φ ( e − 15 π ) = π 4 Γ ( 3 4 ) 7 + 3 3 + 5 + 15 + 60 4 + 1500 4 12 3 8 ⋅ 5 2 φ ( e − 16 π ) = φ ( e − 4 π ) + π 4 Γ ( 3 4 ) 1 + 2 4 128 16 φ ( e − 17 π ) = π 4 Γ ( 3 4 ) 2 ( 1 + 17 4 ) + 17 8 5 + 17 17 + 17 17 2 φ ( e − 20 π ) = φ ( e − 5 π ) + π 4 Γ ( 3 4 ) 3 + 2 5 4 5 2 6 φ ( e − 36 π ) = 3 φ ( e − 9 π ) + 2 φ ( e − 4 π ) − φ ( e − π ) + π 4 Γ ( 3 4 ) 2 4 + 18 4 + 216 4 3 {\displaystyle {\begin{aligned}\varphi \left(e^{-\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}={\sqrt {2}}\,\eta \left({\sqrt {-1}}\right)\\\varphi \left(e^{-2\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {2+{\sqrt {2}}}}{2}}\\\varphi \left(e^{-3\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {1+{\sqrt {3}}}}{\sqrt[{8}]{108}}}\\\varphi \left(e^{-4\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {2+{\sqrt[{4}]{8}}}{4}}\\\varphi \left(e^{-5\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt {\frac {2+{\sqrt {5}}}{5}}}\\\varphi \left(e^{-6\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}+{\sqrt[{4}]{9}}}}{\sqrt[{8}]{12^{3}}}}\\\varphi \left(e^{-7\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt {13+{\sqrt {7}}}}+{\sqrt {7+3{\sqrt {7}}}}}}{{\sqrt[{8}]{14^{3}}}\cdot {\sqrt[{16}]{7}}}}\\\varphi \left(e^{-8\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt {2+{\sqrt {2}}}}+{\sqrt[{8}]{128}}}{4}}\\\varphi \left(e^{-9\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {1+{\sqrt[{3}]{2+2{\sqrt {3}}}}}{3}}\\\varphi \left(e^{-10\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{64}}+{\sqrt[{4}]{80}}+{\sqrt[{4}]{81}}+{\sqrt[{4}]{100}}}}{\sqrt[{4}]{200}}}\\\varphi \left(e^{-11\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {11+{\sqrt {11}}+(5+3{\sqrt {3}}+{\sqrt {11}}+{\sqrt {33}}){\sqrt[{3}]{-44+33{\sqrt {3}}}}+(-5+3{\sqrt {3}}-{\sqrt {11}}+{\sqrt {33}}){\sqrt[{3}]{44+33{\sqrt {3}}}}}}{\sqrt[{8}]{52180524}}}\\\varphi \left(e^{-12\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{2}}+{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}+{\sqrt[{4}]{9}}+{\sqrt[{4}]{18}}+{\sqrt[{4}]{24}}}}{2{\sqrt[{8}]{108}}}}\\\varphi \left(e^{-13\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {13+8{\sqrt {13}}+(11-6{\sqrt {3}}+{\sqrt {13}}){\sqrt[{3}]{143+78{\sqrt {3}}}}+(11+6{\sqrt {3}}+{\sqrt {13}}){\sqrt[{3}]{143-78{\sqrt {3}}}}}}{\sqrt[{4}]{19773}}}\\\varphi \left(e^{-14\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt {13+{\sqrt {7}}}}+{\sqrt {7+3{\sqrt {7}}}}+{\sqrt {10+2{\sqrt {7}}}}+{\sqrt[{8}]{28}}{\sqrt {4+{\sqrt {7}}}}}}{\sqrt[{16}]{28^{7}}}}\\\varphi \left(e^{-15\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {7+3{\sqrt {3}}+{\sqrt {5}}+{\sqrt {15}}+{\sqrt[{4}]{60}}+{\sqrt[{4}]{1500}}}}{{\sqrt[{8}]{12^{3}}}\cdot {\sqrt {5}}}}\\2\varphi \left(e^{-16\pi }\right)&=\varphi \left(e^{-4\pi }\right)+{\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{1+{\sqrt {2}}}}{\sqrt[{16}]{128}}}\\\varphi \left(e^{-17\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt {2}}(1+{\sqrt[{4}]{17}})+{\sqrt[{8}]{17}}{\sqrt {5+{\sqrt {17}}}}}{\sqrt {17+17{\sqrt {17}}}}}\\2\varphi \left(e^{-20\pi }\right)&=\varphi \left(e^{-5\pi }\right)+{\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt {\frac {3+2{\sqrt[{4}]{5}}}{5{\sqrt {2}}}}}\\6\varphi \left(e^{-36\pi }\right)&=3\varphi \left(e^{-9\pi }\right)+2\varphi \left(e^{-4\pi }\right)-\varphi \left(e^{-\pi }\right)+{\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt[{3}]{{\sqrt[{4}]{2}}+{\sqrt[{4}]{18}}+{\sqrt[{4}]{216}}}}\end{aligned}}} If the reciprocal of theGelfond constant is raised to the power of the reciprocal of an odd number, then the correspondingϑ 00 {\displaystyle \vartheta _{00}} ϕ {\displaystyle \phi } hyperbolic lemniscatic sine :
φ [ exp ( − 1 5 π ) ] = π 4 Γ ( 3 4 ) − 1 slh ( 1 5 2 ϖ ) slh ( 2 5 2 ϖ ) {\displaystyle \varphi {\bigl [}\exp(-{\tfrac {1}{5}}\pi ){\bigr ]}={\sqrt[{4}]{\pi }}\,{\Gamma \left({\tfrac {3}{4}}\right)}^{-1}\operatorname {slh} {\bigl (}{\tfrac {1}{5}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {2}{5}}{\sqrt {2}}\,\varpi {\bigr )}} φ [ exp ( − 1 7 π ) ] = π 4 Γ ( 3 4 ) − 1 slh ( 1 7 2 ϖ ) slh ( 2 7 2 ϖ ) slh ( 3 7 2 ϖ ) {\displaystyle \varphi {\bigl [}\exp(-{\tfrac {1}{7}}\pi ){\bigr ]}={\sqrt[{4}]{\pi }}\,{\Gamma \left({\tfrac {3}{4}}\right)}^{-1}\operatorname {slh} {\bigl (}{\tfrac {1}{7}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {2}{7}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {3}{7}}{\sqrt {2}}\,\varpi {\bigr )}} φ [ exp ( − 1 9 π ) ] = π 4 Γ ( 3 4 ) − 1 slh ( 1 9 2 ϖ ) slh ( 2 9 2 ϖ ) slh ( 3 9 2 ϖ ) slh ( 4 9 2 ϖ ) {\displaystyle \varphi {\bigl [}\exp(-{\tfrac {1}{9}}\pi ){\bigr ]}={\sqrt[{4}]{\pi }}\,{\Gamma \left({\tfrac {3}{4}}\right)}^{-1}\operatorname {slh} {\bigl (}{\tfrac {1}{9}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {2}{9}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {3}{9}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {4}{9}}{\sqrt {2}}\,\varpi {\bigr )}} φ [ exp ( − 1 11 π ) ] = π 4 Γ ( 3 4 ) − 1 slh ( 1 11 2 ϖ ) slh ( 2 11 2 ϖ ) slh ( 3 11 2 ϖ ) slh ( 4 11 2 ϖ ) slh ( 5 11 2 ϖ ) {\displaystyle \varphi {\bigl [}\exp(-{\tfrac {1}{11}}\pi ){\bigr ]}={\sqrt[{4}]{\pi }}\,{\Gamma \left({\tfrac {3}{4}}\right)}^{-1}\operatorname {slh} {\bigl (}{\tfrac {1}{11}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {2}{11}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {3}{11}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {4}{11}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {5}{11}}{\sqrt {2}}\,\varpi {\bigr )}} With the letterϖ {\displaystyle \varpi } Lemniscate constant is represented.
Note that the following modular identities hold:
2 φ ( q 4 ) = φ ( q ) + 2 φ 2 ( q 2 ) − φ 2 ( q ) 3 φ ( q 9 ) = φ ( q ) + 9 φ 4 ( q 3 ) φ ( q ) − φ 3 ( q ) 3 5 φ ( q 25 ) = φ ( q 5 ) cot ( 1 2 arctan ( 2 5 φ ( q ) φ ( q 5 ) φ 2 ( q ) − φ 2 ( q 5 ) 1 + s ( q ) − s 2 ( q ) s ( q ) ) ) {\displaystyle {\begin{aligned}2\varphi \left(q^{4}\right)&=\varphi (q)+{\sqrt {2\varphi ^{2}\left(q^{2}\right)-\varphi ^{2}(q)}}\\3\varphi \left(q^{9}\right)&=\varphi (q)+{\sqrt[{3}]{9{\frac {\varphi ^{4}\left(q^{3}\right)}{\varphi (q)}}-\varphi ^{3}(q)}}\\{\sqrt {5}}\varphi \left(q^{25}\right)&=\varphi \left(q^{5}\right)\cot \left({\frac {1}{2}}\arctan \left({\frac {2}{\sqrt {5}}}{\frac {\varphi (q)\varphi \left(q^{5}\right)}{\varphi ^{2}(q)-\varphi ^{2}\left(q^{5}\right)}}{\frac {1+s(q)-s^{2}(q)}{s(q)}}\right)\right)\end{aligned}}} wheres ( q ) = s ( e π i τ ) = − R ( − e − π i / ( 5 τ ) ) {\displaystyle s(q)=s\left(e^{\pi i\tau }\right)=-R\left(-e^{-\pi i/(5\tau )}\right)} Rogers–Ramanujan continued fraction :
s ( q ) = tan ( 1 2 arctan ( 5 2 φ 2 ( q 5 ) φ 2 ( q ) − 1 2 ) ) cot 2 ( 1 2 arccot ( 5 2 φ 2 ( q 5 ) φ 2 ( q ) − 1 2 ) ) 5 = e − π i / ( 25 τ ) 1 − e − π i / ( 5 τ ) 1 + e − 2 π i / ( 5 τ ) 1 − ⋱ {\displaystyle {\begin{aligned}s(q)&={\sqrt[{5}]{\tan \left({\frac {1}{2}}\arctan \left({\frac {5}{2}}{\frac {\varphi ^{2}\left(q^{5}\right)}{\varphi ^{2}(q)}}-{\frac {1}{2}}\right)\right)\cot ^{2}\left({\frac {1}{2}}\operatorname {arccot} \left({\frac {5}{2}}{\frac {\varphi ^{2}\left(q^{5}\right)}{\varphi ^{2}(q)}}-{\frac {1}{2}}\right)\right)}}\\&={\cfrac {e^{-\pi i/(25\tau )}}{1-{\cfrac {e^{-\pi i/(5\tau )}}{1+{\cfrac {e^{-2\pi i/(5\tau )}}{1-\ddots }}}}}}\end{aligned}}} The mathematicianBruce Berndt found out further values[ 7]
φ ( exp ( − 3 π ) ) = π − 1 Γ ( 4 3 ) 3 / 2 2 − 2 / 3 3 13 / 8 φ ( exp ( − 2 3 π ) ) = π − 1 Γ ( 4 3 ) 3 / 2 2 − 2 / 3 3 13 / 8 cos ( 1 24 π ) φ ( exp ( − 3 3 π ) ) = π − 1 Γ ( 4 3 ) 3 / 2 2 − 2 / 3 3 7 / 8 ( 2 3 + 1 ) φ ( exp ( − 4 3 π ) ) = π − 1 Γ ( 4 3 ) 3 / 2 2 − 5 / 3 3 13 / 8 ( 1 + cos ( 1 12 π ) ) φ ( exp ( − 5 3 π ) ) = π − 1 Γ ( 4 3 ) 3 / 2 2 − 2 / 3 3 5 / 8 sin ( 1 5 π ) ( 2 5 100 3 + 2 5 10 3 + 3 5 5 + 1 ) {\displaystyle {\begin{array}{lll}\varphi \left(\exp(-{\sqrt {3}}\,\pi )\right)&=&\pi ^{-1}{\Gamma \left({\tfrac {4}{3}}\right)}^{3/2}2^{-2/3}3^{13/8}\\\varphi \left(\exp(-2{\sqrt {3}}\,\pi )\right)&=&\pi ^{-1}{\Gamma \left({\tfrac {4}{3}}\right)}^{3/2}2^{-2/3}3^{13/8}\cos({\tfrac {1}{24}}\pi )\\\varphi \left(\exp(-3{\sqrt {3}}\,\pi )\right)&=&\pi ^{-1}{\Gamma \left({\tfrac {4}{3}}\right)}^{3/2}2^{-2/3}3^{7/8}({\sqrt[{3}]{2}}+1)\\\varphi \left(\exp(-4{\sqrt {3}}\,\pi )\right)&=&\pi ^{-1}{\Gamma \left({\tfrac {4}{3}}\right)}^{3/2}2^{-5/3}3^{13/8}{\Bigl (}1+{\sqrt {\cos({\tfrac {1}{12}}\pi )}}{\Bigr )}\\\varphi \left(\exp(-5{\sqrt {3}}\,\pi )\right)&=&\pi ^{-1}{\Gamma \left({\tfrac {4}{3}}\right)}^{3/2}2^{-2/3}3^{5/8}\sin({\tfrac {1}{5}}\pi )({\tfrac {2}{5}}{\sqrt[{3}]{100}}+{\tfrac {2}{5}}{\sqrt[{3}]{10}}+{\tfrac {3}{5}}{\sqrt {5}}+1)\end{array}}} Many values of the theta function[ 8]
φ ( exp ( − 2 π ) ) = π − 1 / 2 Γ ( 9 8 ) Γ ( 5 4 ) − 1 / 2 2 7 / 8 φ ( exp ( − 2 2 π ) ) = π − 1 / 2 Γ ( 9 8 ) Γ ( 5 4 ) − 1 / 2 2 1 / 8 ( 1 + 2 − 1 ) φ ( exp ( − 3 2 π ) ) = π − 1 / 2 Γ ( 9 8 ) Γ ( 5 4 ) − 1 / 2 2 3 / 8 3 − 1 / 2 ( 3 + 1 ) tan ( 5 24 π ) φ ( exp ( − 4 2 π ) ) = π − 1 / 2 Γ ( 9 8 ) Γ ( 5 4 ) − 1 / 2 2 − 1 / 8 ( 1 + 2 2 − 2 4 ) φ ( exp ( − 5 2 π ) ) = π − 1 / 2 Γ ( 9 8 ) Γ ( 5 4 ) − 1 / 2 1 15 2 3 / 8 × × [ 5 3 10 + 2 5 ( 5 + 2 + 3 3 3 + 5 + 2 − 3 3 3 ) − ( 2 − 2 ) 25 − 10 5 ] φ ( exp ( − 6 π ) ) = π − 1 / 2 Γ ( 5 24 ) Γ ( 5 12 ) − 1 / 2 2 − 13 / 24 3 − 1 / 8 sin ( 5 12 π ) φ ( exp ( − 1 2 6 π ) ) = π − 1 / 2 Γ ( 5 24 ) Γ ( 5 12 ) − 1 / 2 2 5 / 24 3 − 1 / 8 sin ( 5 24 π ) {\displaystyle {\begin{array}{lll}\varphi \left(\exp(-{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{7/8}\\\varphi \left(\exp(-2{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{1/8}{\Bigl (}1+{\sqrt {{\sqrt {2}}-1}}{\Bigr )}\\\varphi \left(\exp(-3{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{3/8}3^{-1/2}({\sqrt {3}}+1){\sqrt {\tan({\tfrac {5}{24}}\pi )}}\\\varphi \left(\exp(-4{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{-1/8}{\Bigl (}1+{\sqrt[{4}]{2{\sqrt {2}}-2}}{\Bigr )}\\\varphi \left(\exp(-5{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}{\frac {1}{15}}\,2^{3/8}\times \\&&\times {\biggl [}{\sqrt[{3}]{5}}\,{\sqrt {10+2{\sqrt {5}}}}{\biggl (}{\sqrt[{3}]{5+{\sqrt {2}}+3{\sqrt {3}}}}+{\sqrt[{3}]{5+{\sqrt {2}}-3{\sqrt {3}}}}\,{\biggr )}-{\bigl (}2-{\sqrt {2}}\,{\bigr )}{\sqrt {25-10{\sqrt {5}}}}\,{\biggr ]}\\\varphi \left(\exp(-{\sqrt {6}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {5}{24}}\right){\Gamma \left({\tfrac {5}{12}}\right)}^{-1/2}2^{-13/24}3^{-1/8}{\sqrt {\sin({\tfrac {5}{12}}\pi )}}\\\varphi \left(\exp(-{\tfrac {1}{2}}{\sqrt {6}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {5}{24}}\right){\Gamma \left({\tfrac {5}{12}}\right)}^{-1/2}2^{5/24}3^{-1/8}\sin({\tfrac {5}{24}}\pi )\end{array}}} Nome power theorems [ edit ] Direct power theorems [ edit ] For the transformation of the nome[ 9]
θ 2 ( q 2 ) = 1 2 2 [ θ 3 ( q ) 2 − θ 4 ( q ) 2 ] {\displaystyle \theta _{2}(q^{2})={\tfrac {1}{2}}{\sqrt {2[\theta _{3}(q)^{2}-\theta _{4}(q)^{2}]}}} θ 3 ( q 2 ) = 1 2 2 [ θ 3 ( q ) 2 + θ 4 ( q ) 2 ] {\displaystyle \theta _{3}(q^{2})={\tfrac {1}{2}}{\sqrt {2[\theta _{3}(q)^{2}+\theta _{4}(q)^{2}]}}} θ 4 ( q 2 ) = θ 4 ( q ) θ 3 ( q ) {\displaystyle \theta _{4}(q^{2})={\sqrt {\theta _{4}(q)\theta _{3}(q)}}} The squares of the three theta zero-value functions with the square function as the inner function are also formed in the pattern of thePythagorean triples according to the Jacobi Identity. Furthermore, those transformations are valid:
θ 3 ( q 4 ) = 1 2 θ 3 ( q ) + 1 2 θ 4 ( q ) {\displaystyle \theta _{3}(q^{4})={\tfrac {1}{2}}\theta _{3}(q)+{\tfrac {1}{2}}\theta _{4}(q)} These formulas can be used to compute the theta values of the cube of the nome:
27 θ 3 ( q 3 ) 8 − 18 θ 3 ( q 3 ) 4 θ 3 ( q ) 4 − θ 3 ( q ) 8 = 8 θ 3 ( q 3 ) 2 θ 3 ( q ) 2 [ 2 θ 4 ( q ) 4 − θ 3 ( q ) 4 ] {\displaystyle 27\,\theta _{3}(q^{3})^{8}-18\,\theta _{3}(q^{3})^{4}\theta _{3}(q)^{4}-\,\theta _{3}(q)^{8}=8\,\theta _{3}(q^{3})^{2}\theta _{3}(q)^{2}[2\,\theta _{4}(q)^{4}-\theta _{3}(q)^{4}]} 27 θ 4 ( q 3 ) 8 − 18 θ 4 ( q 3 ) 4 θ 4 ( q ) 4 − θ 4 ( q ) 8 = 8 θ 4 ( q 3 ) 2 θ 4 ( q ) 2 [ 2 θ 3 ( q ) 4 − θ 4 ( q ) 4 ] {\displaystyle 27\,\theta _{4}(q^{3})^{8}-18\,\theta _{4}(q^{3})^{4}\theta _{4}(q)^{4}-\,\theta _{4}(q)^{8}=8\,\theta _{4}(q^{3})^{2}\theta _{4}(q)^{2}[2\,\theta _{3}(q)^{4}-\theta _{4}(q)^{4}]} And the following formulas can be used to compute the theta values of the fifth power of the nome:
[ θ 3 ( q ) 2 − θ 3 ( q 5 ) 2 ] [ 5 θ 3 ( q 5 ) 2 − θ 3 ( q ) 2 ] 5 = 256 θ 3 ( q 5 ) 2 θ 3 ( q ) 2 θ 4 ( q ) 4 [ θ 3 ( q ) 4 − θ 4 ( q ) 4 ] {\displaystyle [\theta _{3}(q)^{2}-\theta _{3}(q^{5})^{2}][5\,\theta _{3}(q^{5})^{2}-\theta _{3}(q)^{2}]^{5}=256\,\theta _{3}(q^{5})^{2}\theta _{3}(q)^{2}\theta _{4}(q)^{4}[\theta _{3}(q)^{4}-\theta _{4}(q)^{4}]} [ θ 4 ( q 5 ) 2 − θ 4 ( q ) 2 ] [ 5 θ 4 ( q 5 ) 2 − θ 4 ( q ) 2 ] 5 = 256 θ 4 ( q 5 ) 2 θ 4 ( q ) 2 θ 3 ( q ) 4 [ θ 3 ( q ) 4 − θ 4 ( q ) 4 ] {\displaystyle [\theta _{4}(q^{5})^{2}-\theta _{4}(q)^{2}][5\,\theta _{4}(q^{5})^{2}-\theta _{4}(q)^{2}]^{5}=256\,\theta _{4}(q^{5})^{2}\theta _{4}(q)^{2}\theta _{3}(q)^{4}[\theta _{3}(q)^{4}-\theta _{4}(q)^{4}]} Transformation at the cube root of the nome [ edit ] The formulas for the theta Nullwert function values from the cube root of the elliptic nome are obtained by contrasting the two real solutions of the corresponding quartic equations:
[ θ 3 ( q 1 / 3 ) 2 θ 3 ( q ) 2 − 3 θ 3 ( q 3 ) 2 θ 3 ( q ) 2 ] 2 = 4 − 4 [ 2 θ 2 ( q ) 2 θ 4 ( q ) 2 θ 3 ( q ) 4 ] 2 / 3 {\displaystyle {\biggl [}{\frac {\theta _{3}(q^{1/3})^{2}}{\theta _{3}(q)^{2}}}-{\frac {3\,\theta _{3}(q^{3})^{2}}{\theta _{3}(q)^{2}}}{\biggr ]}^{2}=4-4{\biggl [}{\frac {2\,\theta _{2}(q)^{2}\theta _{4}(q)^{2}}{\theta _{3}(q)^{4}}}{\biggr ]}^{2/3}} [ 3 θ 4 ( q 3 ) 2 θ 4 ( q ) 2 − θ 4 ( q 1 / 3 ) 2 θ 4 ( q ) 2 ] 2 = 4 + 4 [ 2 θ 2 ( q ) 2 θ 3 ( q ) 2 θ 4 ( q ) 4 ] 2 / 3 {\displaystyle {\biggl [}{\frac {3\,\theta _{4}(q^{3})^{2}}{\theta _{4}(q)^{2}}}-{\frac {\theta _{4}(q^{1/3})^{2}}{\theta _{4}(q)^{2}}}{\biggr ]}^{2}=4+4{\biggl [}{\frac {2\,\theta _{2}(q)^{2}\theta _{3}(q)^{2}}{\theta _{4}(q)^{4}}}{\biggr ]}^{2/3}} Transformation at the fifth root of the nome [ edit ] TheRogers-Ramanujan continued fraction can be defined in terms of theJacobi theta function in the following way:
R ( q ) = tan { 1 2 arctan [ 1 2 − θ 4 ( q ) 2 2 θ 4 ( q 5 ) 2 ] } 1 / 5 tan { 1 2 arccot [ 1 2 − θ 4 ( q ) 2 2 θ 4 ( q 5 ) 2 ] } 2 / 5 {\displaystyle R(q)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(q)^{2}}{2\,\theta _{4}(q^{5})^{2}}}{\biggr ]}{\biggr \}}^{1/5}\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(q)^{2}}{2\,\theta _{4}(q^{5})^{2}}}{\biggr ]}{\biggr \}}^{2/5}} R ( q 2 ) = tan { 1 2 arctan [ 1 2 − θ 4 ( q ) 2 2 θ 4 ( q 5 ) 2 ] } 2 / 5 cot { 1 2 arccot [ 1 2 − θ 4 ( q ) 2 2 θ 4 ( q 5 ) 2 ] } 1 / 5 {\displaystyle R(q^{2})=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(q)^{2}}{2\,\theta _{4}(q^{5})^{2}}}{\biggr ]}{\biggr \}}^{2/5}\cot {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(q)^{2}}{2\,\theta _{4}(q^{5})^{2}}}{\biggr ]}{\biggr \}}^{1/5}} R ( q 2 ) = tan { 1 2 arctan [ θ 3 ( q ) 2 2 θ 3 ( q 5 ) 2 − 1 2 ] } 2 / 5 tan { 1 2 arccot [ θ 3 ( q ) 2 2 θ 3 ( q 5 ) 2 − 1 2 ] } 1 / 5 {\displaystyle R(q^{2})=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {\theta _{3}(q)^{2}}{2\,\theta _{3}(q^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{2/5}\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\theta _{3}(q)^{2}}{2\,\theta _{3}(q^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{1/5}} The alternating Rogers-Ramanujan continued fraction function S(q) has the following two identities:
S ( q ) = R ( q 4 ) R ( q 2 ) R ( q ) = tan { 1 2 arctan [ θ 3 ( q ) 2 2 θ 3 ( q 5 ) 2 − 1 2 ] } 1 / 5 cot { 1 2 arccot [ θ 3 ( q ) 2 2 θ 3 ( q 5 ) 2 − 1 2 ] } 2 / 5 {\displaystyle S(q)={\frac {R(q^{4})}{R(q^{2})R(q)}}=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {\theta _{3}(q)^{2}}{2\,\theta _{3}(q^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{1/5}\cot {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\theta _{3}(q)^{2}}{2\,\theta _{3}(q^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{2/5}} The theta function values from the fifth root of the nome can be represented as a rational combination of the continued fractions R and S and the theta function values from the fifth power of the nome and the nome itself. The following four equations are valid for all values q between 0 and 1:
θ 3 ( q 1 / 5 ) θ 3 ( q 5 ) − 1 = 1 S ( q ) [ S ( q ) 2 + R ( q 2 ) ] [ 1 + R ( q 2 ) S ( q ) ] {\displaystyle {\frac {\theta _{3}(q^{1/5})}{\theta _{3}(q^{5})}}-1={\frac {1}{S(q)}}{\bigl [}S(q)^{2}+R(q^{2}){\bigr ]}{\bigl [}1+R(q^{2})S(q){\bigr ]}} 1 − θ 4 ( q 1 / 5 ) θ 4 ( q 5 ) = 1 R ( q ) [ R ( q 2 ) + R ( q ) 2 ] [ 1 − R ( q 2 ) R ( q ) ] {\displaystyle 1-{\frac {\theta _{4}(q^{1/5})}{\theta _{4}(q^{5})}}={\frac {1}{R(q)}}{\bigl [}R(q^{2})+R(q)^{2}{\bigr ]}{\bigl [}1-R(q^{2})R(q){\bigr ]}} θ 3 ( q 1 / 5 ) 2 − θ 3 ( q ) 2 = [ θ 3 ( q ) 2 − θ 3 ( q 5 ) 2 ] [ 1 + 1 R ( q 2 ) S ( q ) + R ( q 2 ) S ( q ) + 1 R ( q 2 ) 2 + R ( q 2 ) 2 + 1 S ( q ) − S ( q ) ] {\displaystyle \theta _{3}(q^{1/5})^{2}-\theta _{3}(q)^{2}={\bigl [}\theta _{3}(q)^{2}-\theta _{3}(q^{5})^{2}{\bigr ]}{\biggl [}1+{\frac {1}{R(q^{2})S(q)}}+R(q^{2})S(q)+{\frac {1}{R(q^{2})^{2}}}+R(q^{2})^{2}+{\frac {1}{S(q)}}-S(q){\biggr ]}} θ 4 ( q ) 2 − θ 4 ( q 1 / 5 ) 2 = [ θ 4 ( q 5 ) 2 − θ 4 ( q ) 2 ] [ 1 − 1 R ( q 2 ) R ( q ) − R ( q 2 ) R ( q ) + 1 R ( q 2 ) 2 + R ( q 2 ) 2 − 1 R ( q ) + R ( q ) ] {\displaystyle \theta _{4}(q)^{2}-\theta _{4}(q^{1/5})^{2}={\bigl [}\theta _{4}(q^{5})^{2}-\theta _{4}(q)^{2}{\bigr ]}{\biggl [}1-{\frac {1}{R(q^{2})R(q)}}-R(q^{2})R(q)+{\frac {1}{R(q^{2})^{2}}}+R(q^{2})^{2}-{\frac {1}{R(q)}}+R(q){\biggr ]}} Modulus dependent theorems [ edit ] In combination with the elliptic modulus, the following formulas can be displayed:
These are the formulas for the square of the elliptic nome:
θ 4 [ q ( k ) ] = θ 4 [ q ( k ) 2 ] 1 − k 2 8 {\displaystyle \theta _{4}[q(k)]=\theta _{4}[q(k)^{2}]{\sqrt[{8}]{1-k^{2}}}} θ 4 [ q ( k ) 2 ] = θ 3 [ q ( k ) ] 1 − k 2 8 {\displaystyle \theta _{4}[q(k)^{2}]=\theta _{3}[q(k)]{\sqrt[{8}]{1-k^{2}}}} θ 3 [ q ( k ) 2 ] = θ 3 [ q ( k ) ] cos [ 1 2 arcsin ( k ) ] {\displaystyle \theta _{3}[q(k)^{2}]=\theta _{3}[q(k)]\cos[{\tfrac {1}{2}}\arcsin(k)]} And this is an efficient formula for the cube of the nome:
θ 4 ⟨ q { tan [ 1 2 arctan ( t 3 ) ] } 3 ⟩ = θ 4 ⟨ q { tan [ 1 2 arctan ( t 3 ) ] } ⟩ 3 − 1 / 2 ( 2 t 4 − t 2 + 1 − t 2 + 2 + t 2 + 1 ) 1 / 2 {\displaystyle \theta _{4}{\biggl \langle }q{\bigl \{}\tan {\bigl [}{\tfrac {1}{2}}\arctan(t^{3}){\bigr ]}{\bigr \}}^{3}{\biggr \rangle }=\theta _{4}{\biggl \langle }q{\bigl \{}\tan {\bigl [}{\tfrac {1}{2}}\arctan(t^{3}){\bigr ]}{\bigr \}}{\biggr \rangle }\,3^{-1/2}{\bigl (}{\sqrt {2{\sqrt {t^{4}-t^{2}+1}}-t^{2}+2}}+{\sqrt {t^{2}+1}}\,{\bigr )}^{1/2}} For all real valuest ∈ R {\displaystyle t\in \mathbb {R} }
And for this formula two examples shall be given:
First calculation example with the valuet = 1 {\displaystyle t=1}
Second calculation example with the valuet = Φ − 2 {\displaystyle t=\Phi ^{-2}}
The constantΦ {\displaystyle \Phi } Golden ratio numberΦ = 1 2 ( 5 + 1 ) {\displaystyle \Phi ={\tfrac {1}{2}}({\sqrt {5}}+1)}
Some series identities [ edit ] Sums with theta function in the result [ edit ] The infinite sum[ 10] [ 11] Fibonacci numbers with odd indices has the identity:
∑ n = 1 ∞ 1 F 2 n − 1 = 5 2 ∑ n = 1 ∞ 2 ( Φ − 2 ) n − 1 / 2 1 + ( Φ − 2 ) 2 n − 1 = 5 4 ∑ a = − ∞ ∞ 2 ( Φ − 2 ) a − 1 / 2 1 + ( Φ − 2 ) 2 a − 1 = {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}={\frac {\sqrt {5}}{2}}\,\sum _{n=1}^{\infty }{\frac {2(\Phi ^{-2})^{n-1/2}}{1+(\Phi ^{-2})^{2n-1}}}={\frac {\sqrt {5}}{4}}\sum _{a=-\infty }^{\infty }{\frac {2(\Phi ^{-2})^{a-1/2}}{1+(\Phi ^{-2})^{2a-1}}}=} = 5 4 θ 2 ( Φ − 2 ) 2 = 5 8 [ θ 3 ( Φ − 1 ) 2 − θ 4 ( Φ − 1 ) 2 ] {\displaystyle ={\frac {\sqrt {5}}{4}}\,\theta _{2}(\Phi ^{-2})^{2}={\frac {\sqrt {5}}{8}}{\bigl [}\theta _{3}(\Phi ^{-1})^{2}-\theta _{4}(\Phi ^{-1})^{2}{\bigr ]}} By not using the theta function expression, following identity between two sums can be formulated:
∑ n = 1 ∞ 1 F 2 n − 1 = 5 4 [ ∑ n = 1 ∞ 2 Φ − ( 2 n − 1 ) 2 / 2 ] 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}={\frac {\sqrt {5}}{4}}\,{\biggl [}\sum _{n=1}^{\infty }2\,\Phi ^{-(2n-1)^{2}/2}{\biggr ]}^{2}} ∑ n = 1 ∞ 1 F 2 n − 1 = 1.82451515740692456814215840626732817332 … {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}=1.82451515740692456814215840626732817332\ldots } Also in this caseΦ = 1 2 ( 5 + 1 ) {\displaystyle \Phi ={\tfrac {1}{2}}({\sqrt {5}}+1)} Golden ratio number again.
Infinite sum of the reciprocals of the Fibonacci number squares:
∑ n = 1 ∞ 1 F n 2 = 5 24 [ 2 θ 2 ( Φ − 2 ) 4 − θ 3 ( Φ − 2 ) 4 + 1 ] = 5 24 [ θ 3 ( Φ − 2 ) 4 − 2 θ 4 ( Φ − 2 ) 4 + 1 ] {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{n}^{2}}}={\frac {5}{24}}{\bigl [}2\,\theta _{2}(\Phi ^{-2})^{4}-\theta _{3}(\Phi ^{-2})^{4}+1{\bigr ]}={\frac {5}{24}}{\bigl [}\theta _{3}(\Phi ^{-2})^{4}-2\,\theta _{4}(\Phi ^{-2})^{4}+1{\bigr ]}} Infinite sum of the reciprocals of thePell numbers with odd indices:
∑ n = 1 ∞ 1 P 2 n − 1 = 1 2 θ 2 [ ( 2 − 1 ) 2 ] 2 = 1 2 2 [ θ 3 ( 2 − 1 ) 2 − θ 4 ( 2 − 1 ) 2 ] {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P_{2n-1}}}={\frac {1}{\sqrt {2}}}\,\theta _{2}{\bigl [}({\sqrt {2}}-1)^{2}{\bigr ]}^{2}={\frac {1}{2{\sqrt {2}}}}{\bigl [}\theta _{3}({\sqrt {2}}-1)^{2}-\theta _{4}({\sqrt {2}}-1)^{2}{\bigr ]}} Sums with theta function in the summand [ edit ] The next two series identities were proved byIstván Mező :[ 12]
θ 4 2 ( q ) = i q 1 4 ∑ k = − ∞ ∞ q 2 k 2 − k θ 1 ( 2 k − 1 2 i ln q , q ) , θ 4 2 ( q ) = ∑ k = − ∞ ∞ q 2 k 2 θ 4 ( k ln q i , q ) . {\displaystyle {\begin{aligned}\theta _{4}^{2}(q)&=iq^{\frac {1}{4}}\sum _{k=-\infty }^{\infty }q^{2k^{2}-k}\theta _{1}\left({\frac {2k-1}{2i}}\ln q,q\right),\\[6pt]\theta _{4}^{2}(q)&=\sum _{k=-\infty }^{\infty }q^{2k^{2}}\theta _{4}\left({\frac {k\ln q}{i}},q\right).\end{aligned}}} These relations hold for all0 <q < 1 . Specializing the values ofq , we have the next parameter free sums
π e π 2 ⋅ 1 Γ 2 ( 3 4 ) = i ∑ k = − ∞ ∞ e π ( k − 2 k 2 ) θ 1 ( i π 2 ( 2 k − 1 ) , e − π ) {\displaystyle {\sqrt {\frac {\pi {\sqrt {e^{\pi }}}}{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}=i\sum _{k=-\infty }^{\infty }e^{\pi \left(k-2k^{2}\right)}\theta _{1}\left({\frac {i\pi }{2}}(2k-1),e^{-\pi }\right)} π 2 ⋅ 1 Γ 2 ( 3 4 ) = ∑ k = − ∞ ∞ θ 4 ( i k π , e − π ) e 2 π k 2 {\displaystyle {\sqrt {\frac {\pi }{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}=\sum _{k=-\infty }^{\infty }{\frac {\theta _{4}\left(ik\pi ,e^{-\pi }\right)}{e^{2\pi k^{2}}}}} Zeros of the Jacobi theta functions [ edit ] All zeros of the Jacobi theta functions are simple zeros and are given by the following:
ϑ ( z ; τ ) = ϑ 00 ( z ; τ ) = 0 ⟺ z = m + n τ + 1 2 + τ 2 ϑ 11 ( z ; τ ) = 0 ⟺ z = m + n τ ϑ 10 ( z ; τ ) = 0 ⟺ z = m + n τ + 1 2 ϑ 01 ( z ; τ ) = 0 ⟺ z = m + n τ + τ 2 {\displaystyle {\begin{aligned}\vartheta (z;\tau )=\vartheta _{00}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {1}{2}}+{\frac {\tau }{2}}\\[3pt]\vartheta _{11}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau \\[3pt]\vartheta _{10}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {1}{2}}\\[3pt]\vartheta _{01}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {\tau }{2}}\end{aligned}}} wherem ,n are arbitrary integers.
Relation to the Riemann zeta function [ edit ] The relation
ϑ ( 0 ; − 1 τ ) = ( − i τ ) 1 2 ϑ ( 0 ; τ ) {\displaystyle \vartheta \left(0;-{\frac {1}{\tau }}\right)=\left(-i\tau \right)^{\frac {1}{2}}\vartheta (0;\tau )} was used byRiemann to prove the functional equation for theRiemann zeta function , by means of theMellin transform
Γ ( s 2 ) π − s 2 ζ ( s ) = 1 2 ∫ 0 ∞ ( ϑ ( 0 ; i t ) − 1 ) t s 2 d t t {\displaystyle \Gamma \left({\frac {s}{2}}\right)\pi ^{-{\frac {s}{2}}}\zeta (s)={\frac {1}{2}}\int _{0}^{\infty }{\bigl (}\vartheta (0;it)-1{\bigr )}t^{\frac {s}{2}}{\frac {\mathrm {d} t}{t}}} which can be shown to be invariant under substitution ofs by1 −s . The corresponding integral forz ≠ 0Hurwitz zeta function .
Relation to the Weierstrass elliptic function [ edit ] The theta function was used by Jacobi to construct (in a form adapted to easy calculation)his elliptic functions as the quotients of the above four theta functions, and could have been used by him to constructWeierstrass's elliptic functions also, since
℘ ( z ; τ ) = − ( log ϑ 11 ( z ; τ ) ) ″ + c {\displaystyle \wp (z;\tau )=-{\big (}\log \vartheta _{11}(z;\tau ){\big )}''+c} where the second derivative is with respect toz and the constantc is defined so that theLaurent expansion of℘(z ) atz = 0
Relation to theq -gamma function [ edit ] The fourth theta function – and thus the others too – is intimately connected to theJacksonq -gamma function via the relation[ 13]
( Γ q 2 ( x ) Γ q 2 ( 1 − x ) ) − 1 = q 2 x ( 1 − x ) ( q − 2 ; q − 2 ) ∞ 3 ( q 2 − 1 ) θ 4 ( 1 2 i ( 1 − 2 x ) log q , 1 q ) . {\displaystyle \left(\Gamma _{q^{2}}(x)\Gamma _{q^{2}}(1-x)\right)^{-1}={\frac {q^{2x(1-x)}}{\left(q^{-2};q^{-2}\right)_{\infty }^{3}\left(q^{2}-1\right)}}\theta _{4}\left({\frac {1}{2i}}(1-2x)\log q,{\frac {1}{q}}\right).} Relations to Dedekind eta function [ edit ] Letη (τ )Dedekind eta function , and the argument of the theta function as thenome q =e πiτ
θ 2 ( q ) = ϑ 10 ( 0 ; τ ) = 2 η 2 ( 2 τ ) η ( τ ) , θ 3 ( q ) = ϑ 00 ( 0 ; τ ) = η 5 ( τ ) η 2 ( 1 2 τ ) η 2 ( 2 τ ) = η 2 ( 1 2 ( τ + 1 ) ) η ( τ + 1 ) , θ 4 ( q ) = ϑ 01 ( 0 ; τ ) = η 2 ( 1 2 τ ) η ( τ ) , {\displaystyle {\begin{aligned}\theta _{2}(q)=\vartheta _{10}(0;\tau )&={\frac {2\eta ^{2}(2\tau )}{\eta (\tau )}},\\[3pt]\theta _{3}(q)=\vartheta _{00}(0;\tau )&={\frac {\eta ^{5}(\tau )}{\eta ^{2}\left({\frac {1}{2}}\tau \right)\eta ^{2}(2\tau )}}={\frac {\eta ^{2}\left({\frac {1}{2}}(\tau +1)\right)}{\eta (\tau +1)}},\\[3pt]\theta _{4}(q)=\vartheta _{01}(0;\tau )&={\frac {\eta ^{2}\left({\frac {1}{2}}\tau \right)}{\eta (\tau )}},\end{aligned}}} and,
θ 2 ( q ) θ 3 ( q ) θ 4 ( q ) = 2 η 3 ( τ ) . {\displaystyle \theta _{2}(q)\,\theta _{3}(q)\,\theta _{4}(q)=2\eta ^{3}(\tau ).} See also theWeber modular functions .
Theelliptic modulus is
k ( τ ) = ϑ 10 ( 0 ; τ ) 2 ϑ 00 ( 0 ; τ ) 2 {\displaystyle k(\tau )={\frac {\vartheta _{10}(0;\tau )^{2}}{\vartheta _{00}(0;\tau )^{2}}}} and the complementary elliptic modulus is
k ′ ( τ ) = ϑ 01 ( 0 ; τ ) 2 ϑ 00 ( 0 ; τ ) 2 {\displaystyle k'(\tau )={\frac {\vartheta _{01}(0;\tau )^{2}}{\vartheta _{00}(0;\tau )^{2}}}} Derivatives of theta functions [ edit ] These are two identical definitions of the complete elliptic integral of the second kind:
E ( k ) = ∫ 0 π / 2 1 − k 2 sin ( φ ) 2 d φ {\displaystyle E(k)=\int _{0}^{\pi /2}{\sqrt {1-k^{2}\sin(\varphi )^{2}}}d\varphi } E ( k ) = π 2 ∑ a = 0 ∞ [ ( 2 a ) ! ] 2 ( 1 − 2 a ) 16 a ( a ! ) 4 k 2 a {\displaystyle E(k)={\frac {\pi }{2}}\sum _{a=0}^{\infty }{\frac {[(2a)!]^{2}}{(1-2a)16^{a}(a!)^{4}}}k^{2a}} The derivatives of the Theta Nullwert functions have these MacLaurin series:
θ 2 ′ ( x ) = d d x θ 2 ( x ) = 1 2 x − 3 / 4 + ∑ n = 1 ∞ 1 2 ( 2 n + 1 ) 2 x ( 2 n − 1 ) ( 2 n + 3 ) / 4 {\displaystyle \theta _{2}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{2}(x)={\frac {1}{2}}x^{-3/4}+\sum _{n=1}^{\infty }{\frac {1}{2}}(2n+1)^{2}x^{(2n-1)(2n+3)/4}} θ 3 ′ ( x ) = d d x θ 3 ( x ) = 2 + ∑ n = 1 ∞ 2 ( n + 1 ) 2 x n ( n + 2 ) {\displaystyle \theta _{3}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{3}(x)=2+\sum _{n=1}^{\infty }2(n+1)^{2}x^{n(n+2)}} θ 4 ′ ( x ) = d d x θ 4 ( x ) = − 2 + ∑ n = 1 ∞ 2 ( n + 1 ) 2 ( − 1 ) n + 1 x n ( n + 2 ) {\displaystyle \theta _{4}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{4}(x)=-2+\sum _{n=1}^{\infty }2(n+1)^{2}(-1)^{n+1}x^{n(n+2)}} The derivatives of theta zero-value functions[ 14]
θ 2 ′ ( x ) = d d x θ 2 ( x ) = 1 2 π x θ 2 ( x ) θ 3 ( x ) 2 E [ θ 2 ( x ) 2 θ 3 ( x ) 2 ] {\displaystyle \theta _{2}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{2}(x)={\frac {1}{2\pi x}}\theta _{2}(x)\theta _{3}(x)^{2}E{\biggl [}{\frac {\theta _{2}(x)^{2}}{\theta _{3}(x)^{2}}}{\biggr ]}} θ 3 ′ ( x ) = d d x θ 3 ( x ) = θ 3 ( x ) [ θ 3 ( x ) 2 + θ 4 ( x ) 2 ] { 1 2 π x E [ θ 3 ( x ) 2 − θ 4 ( x ) 2 θ 3 ( x ) 2 + θ 4 ( x ) 2 ] − θ 4 ( x ) 2 4 x } {\displaystyle \theta _{3}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{3}(x)=\theta _{3}(x){\bigl [}\theta _{3}(x)^{2}+\theta _{4}(x)^{2}{\bigr ]}{\biggl \{}{\frac {1}{2\pi x}}E{\biggl [}{\frac {\theta _{3}(x)^{2}-\theta _{4}(x)^{2}}{\theta _{3}(x)^{2}+\theta _{4}(x)^{2}}}{\biggr ]}-{\frac {\theta _{4}(x)^{2}}{4\,x}}{\biggr \}}} θ 4 ′ ( x ) = d d x θ 4 ( x ) = θ 4 ( x ) [ θ 3 ( x ) 2 + θ 4 ( x ) 2 ] { 1 2 π x E [ θ 3 ( x ) 2 − θ 4 ( x ) 2 θ 3 ( x ) 2 + θ 4 ( x ) 2 ] − θ 3 ( x ) 2 4 x } {\displaystyle \theta _{4}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{4}(x)=\theta _{4}(x){\bigl [}\theta _{3}(x)^{2}+\theta _{4}(x)^{2}{\bigr ]}{\biggl \{}{\frac {1}{2\pi x}}E{\biggl [}{\frac {\theta _{3}(x)^{2}-\theta _{4}(x)^{2}}{\theta _{3}(x)^{2}+\theta _{4}(x)^{2}}}{\biggr ]}-{\frac {\theta _{3}(x)^{2}}{4\,x}}{\biggr \}}} The two last mentioned formulas are valid for all real numbers of the real definition interval:− 1 < x < 1 ∩ x ∈ R {\displaystyle -1<x<1\,\cap \,x\in \mathbb {R} }
And these two last named theta derivative functions are related to each other in this way:
ϑ 4 ( x ) [ d d x ϑ 3 ( x ) ] − ϑ 3 ( x ) [ d d x θ 4 ( x ) ] = 1 4 x θ 3 ( x ) θ 4 ( x ) [ θ 3 ( x ) 4 − θ 4 ( x ) 4 ] {\displaystyle \vartheta _{4}(x){\biggl [}{\frac {\mathrm {d} }{\mathrm {d} x}}\,\vartheta _{3}(x){\biggr ]}-\vartheta _{3}(x){\biggl [}{\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{4}(x){\biggr ]}={\frac {1}{4\,x}}\,\theta _{3}(x)\,\theta _{4}(x){\bigl [}\theta _{3}(x)^{4}-\theta _{4}(x)^{4}{\bigr ]}} The derivatives of the quotients from two of the three theta functions mentioned here always have a rational relationship to those three functions:
d d x θ 2 ( x ) θ 3 ( x ) = θ 2 ( x ) θ 4 ( x ) 4 4 x θ 3 ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\,{\frac {\theta _{2}(x)}{\theta _{3}(x)}}={\frac {\theta _{2}(x)\,\theta _{4}(x)^{4}}{4\,x\,\theta _{3}(x)}}} d d x θ 2 ( x ) θ 4 ( x ) = θ 2 ( x ) θ 3 ( x ) 4 4 x θ 4 ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\,{\frac {\theta _{2}(x)}{\theta _{4}(x)}}={\frac {\theta _{2}(x)\,\theta _{3}(x)^{4}}{4\,x\,\theta _{4}(x)}}} d d x θ 3 ( x ) θ 4 ( x ) = θ 3 ( x ) 5 − θ 3 ( x ) θ 4 ( x ) 4 4 x θ 4 ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\,{\frac {\theta _{3}(x)}{\theta _{4}(x)}}={\frac {\theta _{3}(x)^{5}-\theta _{3}(x)\,\theta _{4}(x)^{4}}{4\,x\,\theta _{4}(x)}}} For the derivation of these derivation formulas see the articlesNome (mathematics) andModular lambda function !
Integrals of theta functions [ edit ] For the theta functions these integrals[ 15]
∫ 0 1 θ 2 ( x ) d x = ∑ k = − ∞ ∞ 4 ( 2 k + 1 ) 2 + 4 = π tanh ( π ) ≈ 3.129881 {\displaystyle \int _{0}^{1}\theta _{2}(x)\,\mathrm {d} x=\sum _{k=-\infty }^{\infty }{\frac {4}{(2k+1)^{2}+4}}=\pi \tanh(\pi )\approx 3.129881} ∫ 0 1 θ 3 ( x ) d x = ∑ k = − ∞ ∞ 1 k 2 + 1 = π coth ( π ) ≈ 3.153348 {\displaystyle \int _{0}^{1}\theta _{3}(x)\,\mathrm {d} x=\sum _{k=-\infty }^{\infty }{\frac {1}{k^{2}+1}}=\pi \coth(\pi )\approx 3.153348} ∫ 0 1 θ 4 ( x ) d x = ∑ k = − ∞ ∞ ( − 1 ) k k 2 + 1 = π csch ( π ) ≈ 0.272029 {\displaystyle \int _{0}^{1}\theta _{4}(x)\,\mathrm {d} x=\sum _{k=-\infty }^{\infty }{\frac {(-1)^{k}}{k^{2}+1}}=\pi \,\operatorname {csch} (\pi )\approx 0.272029} The final results now shown are based on the general Cauchy sum formulas.
A solution to the heat equation [ edit ] The Jacobi theta function is thefundamental solution of the one-dimensionalheat equation with spatially periodic boundary conditions.[ 16] z =x τ =it t real and positive, we can write
ϑ ( x ; i t ) = 1 + 2 ∑ n = 1 ∞ exp ( − π n 2 t ) cos ( 2 π n x ) {\displaystyle \vartheta (x;it)=1+2\sum _{n=1}^{\infty }\exp \left(-\pi n^{2}t\right)\cos(2\pi nx)} which solves the heat equation
∂ ∂ t ϑ ( x ; i t ) = 1 4 π ∂ 2 ∂ x 2 ϑ ( x ; i t ) . {\displaystyle {\frac {\partial }{\partial t}}\vartheta (x;it)={\frac {1}{4\pi }}{\frac {\partial ^{2}}{\partial x^{2}}}\vartheta (x;it).} This theta-function solution is 1-periodic inx , and ast → 0delta function , orDirac comb , in the sense ofdistributions
lim t → 0 ϑ ( x ; i t ) = ∑ n = − ∞ ∞ δ ( x − n ) {\displaystyle \lim _{t\to 0}\vartheta (x;it)=\sum _{n=-\infty }^{\infty }\delta (x-n)} General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data att = 0
Relation to the Heisenberg group [ edit ] The Jacobi theta function is invariant under the action of a discrete subgroup of theHeisenberg group . This invariance is presented in the article on thetheta representation of the Heisenberg group.
IfF is aquadratic form inn variables, then the theta function associated withF is
θ F ( z ) = ∑ m ∈ Z n e 2 π i z F ( m ) {\displaystyle \theta _{F}(z)=\sum _{m\in \mathbb {Z} ^{n}}e^{2\pi izF(m)}} with the sum extending over thelattice of integersZ n {\displaystyle \mathbb {Z} ^{n}} modular form of weightn / 2 modular group . In the Fourier expansion,
θ ^ F ( z ) = ∑ k = 0 ∞ R F ( k ) e 2 π i k z , {\displaystyle {\hat {\theta }}_{F}(z)=\sum _{k=0}^{\infty }R_{F}(k)e^{2\pi ikz},} the numbersRF (k )representation numbers of the form.
Theta series of a Dirichlet character [ edit ] Forχ a primitiveDirichlet character moduloq andν =1 −χ (−1) / 2
θ χ ( z ) = 1 2 ∑ n = − ∞ ∞ χ ( n ) n ν e 2 i π n 2 z {\displaystyle \theta _{\chi }(z)={\frac {1}{2}}\sum _{n=-\infty }^{\infty }\chi (n)n^{\nu }e^{2i\pi n^{2}z}} is a weight1 / 2 +ν 4q 2 and character
χ ( d ) ( − 1 d ) ν , {\displaystyle \chi (d)\left({\frac {-1}{d}}\right)^{\nu },} which means[ 17]
θ χ ( a z + b c z + d ) = χ ( d ) ( − 1 d ) ν ( θ 1 ( a z + b c z + d ) θ 1 ( z ) ) 1 + 2 ν θ χ ( z ) {\displaystyle \theta _{\chi }\left({\frac {az+b}{cz+d}}\right)=\chi (d)\left({\frac {-1}{d}}\right)^{\nu }\left({\frac {\theta _{1}\left({\frac {az+b}{cz+d}}\right)}{\theta _{1}(z)}}\right)^{1+2\nu }\theta _{\chi }(z)} whenever
a , b , c , d ∈ Z 4 , a d − b c = 1 , c ≡ 0 mod 4 q 2 . {\displaystyle a,b,c,d\in \mathbb {Z} ^{4},ad-bc=1,c\equiv 0{\bmod {4}}q^{2}.} Ramanujan theta function [ edit ] Riemann theta function [ edit ] Let
H n = { F ∈ M ( n , C ) | F = F T , Im F > 0 } {\displaystyle \mathbb {H} _{n}=\left\{F\in M(n,\mathbb {C} )\,{\big |}\,F=F^{\mathsf {T}}\,,\,\operatorname {Im} F>0\right\}} be the set ofsymmetric squarematrices whose imaginary part ispositive definite .H n {\displaystyle \mathbb {H} _{n}} Siegel upper half-space and is the multi-dimensional analog of theupper half-plane . Then -dimensional analogue of themodular group is thesymplectic group Sp(2n ,Z {\displaystyle \mathbb {Z} } ; forn = 1Sp(2,Z {\displaystyle \mathbb {Z} } Z {\displaystyle \mathbb {Z} } . Then -dimensional analogue of thecongruence subgroups is played by
ker { Sp ( 2 n , Z ) → Sp ( 2 n , Z / k Z ) } . {\displaystyle \ker {\big \{}\operatorname {Sp} (2n,\mathbb {Z} )\to \operatorname {Sp} (2n,\mathbb {Z} /k\mathbb {Z} ){\big \}}.} Then, givenτ ∈H n {\displaystyle \mathbb {H} _{n}} Riemann theta function is defined as
θ ( z , τ ) = ∑ m ∈ Z n exp ( 2 π i ( 1 2 m T τ m + m T z ) ) . {\displaystyle \theta (z,\tau )=\sum _{m\in \mathbb {Z} ^{n}}\exp \left(2\pi i\left({\tfrac {1}{2}}m^{\mathsf {T}}\tau m+m^{\mathsf {T}}z\right)\right).} Here,z ∈C n {\displaystyle \mathbb {C} ^{n}} n -dimensional complex vector, and the superscriptT denotes thetranspose . The Jacobi theta function is then a special case, withn = 1τ ∈H {\displaystyle \mathbb {H} } H {\displaystyle \mathbb {H} } upper half-plane . One major application of the Riemann theta function is that it allows one to give explicit formulas for meromorphic functions on compactRiemann surfaces , as well as other auxiliary objects that figure prominently in their function theory, by takingτ to be the period matrix with respect to a canonical basis for its firsthomology group .
The Riemann theta converges absolutely and uniformly on compact subsets ofC n × H n {\displaystyle \mathbb {C} ^{n}\times \mathbb {H} _{n}}
The functional equation is
θ ( z + a + τ b , τ ) = exp ( 2 π i ( − b T z − 1 2 b T τ b ) ) θ ( z , τ ) {\displaystyle \theta (z+a+\tau b,\tau )=\exp \left(2\pi i\left(-b^{\mathsf {T}}z-{\tfrac {1}{2}}b^{\mathsf {T}}\tau b\right)\right)\theta (z,\tau )} which holds for all vectorsa ,b ∈Z n {\displaystyle \mathbb {Z} ^{n}} z ∈C n {\displaystyle \mathbb {C} ^{n}} τ ∈H n {\displaystyle \mathbb {H} _{n}}
ThePoincaré series generalizes the theta series to automorphic forms with respect to arbitraryFuchsian groups .
Derivation of the theta values [ edit ] Identity of the Euler beta function [ edit ] In the following, three important theta function values are to be derived as examples:
This is how theEuler beta function is defined in its reduced form:
β ( x ) = Γ ( x ) 2 Γ ( 2 x ) {\displaystyle \beta (x)={\frac {\Gamma (x)^{2}}{\Gamma (2x)}}} In general, for all natural numbersn ∈ N {\displaystyle n\in \mathbb {N} }
4 − 1 / ( n + 2 ) n + 2 csc ( π n + 2 ) β [ n 2 ( n + 2 ) ] = ∫ 0 ∞ 1 x n + 2 + 1 d x {\displaystyle {\frac {4^{-1/(n+2)}}{n+2}}\csc {\bigl (}{\frac {\pi }{n+2}}{\bigr )}\beta {\biggl [}{\frac {n}{2(n+2)}}{\biggr ]}=\int _{0}^{\infty }{\frac {1}{\sqrt {x^{n+2}+1}}}\,\mathrm {d} x} Exemplary elliptic integrals [ edit ] In the following someElliptic Integral Singular Values [ 18]
And the following function has the following elliptic antiderivative:
1 x 8 + 1 = {\displaystyle {\frac {1}{\sqrt {x^{8}+1}}}=} = d d x 1 4 sec ( π 8 ) F { 2 arctan [ 2 cos ( π / 8 ) x x 4 + 2 x 2 + 1 − x 2 + 1 ] ; 2 2 4 sin ( π 8 ) } + 1 4 sec ( π 8 ) F { arcsin [ 2 cos ( π / 8 ) x x 2 + 1 ] ; tan ( π 8 ) } {\displaystyle ={\frac {\mathrm {d} }{\mathrm {d} x}}\,{\frac {1}{4}}\sec {\bigl (}{\frac {\pi }{8}}{\bigr )}F{\biggl \{}2\arctan {\biggl [}{\frac {2\cos(\pi /8)\,x}{{\sqrt {x^{4}+{\sqrt {2}}\,x^{2}+1}}-x^{2}+1}}{\biggr ]};2{\sqrt[{4}]{2}}\sin {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr \}}+{\frac {1}{4}}\sec {\bigl (}{\frac {\pi }{8}}{\bigr )}F{\biggl \{}\arcsin {\biggl [}{\frac {2\cos(\pi /8)\,x}{x^{2}+1}}{\biggr ]};\tan {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr \}}} For the valuen = 6 {\displaystyle n=6}
1 8 2 4 csc ( π 8 ) β ( 3 8 ) = ∫ 0 ∞ 1 x 8 + 1 d x = {\displaystyle {\frac {1}{8{\sqrt[{4}]{2}}}}\csc {\bigl (}{\frac {\pi }{8}}{\bigr )}\beta {\bigl (}{\frac {3}{8}}{\bigr )}=\int _{0}^{\infty }{\frac {1}{\sqrt {x^{8}+1}}}\,\mathrm {d} x=} = ⟨ 1 4 sec ( π 8 ) F { 2 arctan [ 2 cos ( π / 8 ) x x 4 + 2 x 2 + 1 − x 2 + 1 ] ; 2 2 4 sin ( π 8 ) } + 1 4 sec ( π 8 ) F { arcsin [ 2 cos ( π / 8 ) x x 2 + 1 ] ; tan ( π 8 ) } ⟩ x = 0 x = ∞ = {\displaystyle ={\biggl \langle }{\color {blue}{\frac {1}{4}}\sec {\bigl (}{\frac {\pi }{8}}{\bigr )}F{\biggl \{}2\arctan {\biggl [}{\frac {2\cos(\pi /8)\,x}{{\sqrt {x^{4}+{\sqrt {2}}\,x^{2}+1}}-x^{2}+1}}{\biggr ]};2{\sqrt[{4}]{2}}\sin {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr \}}+{\frac {1}{4}}\sec {\bigl (}{\frac {\pi }{8}}{\bigr )}F{\biggl \{}\arcsin {\biggl [}{\frac {2\cos(\pi /8)\,x}{x^{2}+1}}{\biggr ]};\tan {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr \}}}{\biggr \rangle }_{x=0}^{x=\infty }=} = 1 4 sec ( π 8 ) F [ π ; 2 2 4 sin ( π 8 ) ] = 1 2 sec ( π 8 ) K ( 2 2 − 2 ) = 2 sin ( π 8 ) K ( 2 − 1 ) {\displaystyle ={\frac {1}{4}}\sec {\bigl (}{\frac {\pi }{8}}{\bigr )}F{\bigl [}\pi ;2{\sqrt[{4}]{2}}\sin {\bigl (}{\frac {\pi }{8}}{\bigr )}{\bigr ]}={\frac {1}{2}}\sec {\bigl (}{\frac {\pi }{8}}{\bigr )}K({\sqrt {2{\sqrt {2}}-2}}{\bigr )}=2\sin {\bigl (}{\frac {\pi }{8}}{\bigr )}K({\sqrt {2}}-1)} This result follows from that equation chain:
K ( 2 − 1 ) = 1 8 2 4 ( 2 + 1 ) β ( 3 8 ) {\displaystyle {\color {ForestGreen}K({\sqrt {2}}-1)={\frac {1}{8}}{\sqrt[{4}]{2}}\,({\sqrt {2}}+1)\,\beta {\bigl (}{\frac {3}{8}}{\bigr )}}}
Combination of the integral identities with the nome [ edit ] The elliptic nome function has these important values:
q ( 1 2 2 ) = exp ( − π ) {\displaystyle q({\tfrac {1}{2}}{\sqrt {2}})=\exp(-\pi )} q [ 1 4 ( 6 − 2 ) ] = exp ( − 3 π ) {\displaystyle q[{\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}})]=\exp(-{\sqrt {3}}\,\pi )} q ( 2 − 1 ) = exp ( − 2 π ) {\displaystyle q({\sqrt {2}}-1)=\exp(-{\sqrt {2}}\,\pi )} For the proof of the correctness of these nome values, see the articleNome (mathematics) !
On the basis of these integral identities and the above-mentionedDefinition and identities to the theta functions in the same section of this article, exemplary theta zero values shall be determined now:
θ 3 [ exp ( − π ) ] = θ 3 [ q ( 1 2 2 ) ] = 2 π − 1 K ( 1 2 2 ) = 2 − 1 / 2 π − 1 / 2 β ( 1 4 ) 1 / 2 = 2 − 1 / 4 π 4 Γ ( 3 4 ) − 1 {\displaystyle \theta _{3}[\exp(-\pi )]=\theta _{3}[q({\tfrac {1}{2}}{\sqrt {2}})]={\sqrt {2\pi ^{-1}K({\tfrac {1}{2}}{\sqrt {2}})}}=2^{-1/2}\pi ^{-1/2}\beta ({\tfrac {1}{4}})^{1/2}=2^{-1/4}{\sqrt[{4}]{\pi }}\,{\Gamma {\bigl (}{\tfrac {3}{4}}{\bigr )}}^{-1}} θ 3 [ exp ( − 3 π ) ] = θ 3 { q [ 1 4 ( 6 − 2 ) ] } = 2 π − 1 K [ 1 4 ( 6 − 2 ) ] = 2 − 1 / 6 3 − 1 / 8 π − 1 / 2 β ( 1 3 ) 1 / 2 {\displaystyle \theta _{3}[\exp(-{\sqrt {3}}\,\pi )]=\theta _{3}{\bigl \{}q{\bigl [}{\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}}){\bigr ]}{\bigr \}}={\sqrt {2\pi ^{-1}K{\bigl [}{\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}}){\bigr ]}}}=2^{-1/6}3^{-1/8}\pi ^{-1/2}\beta ({\tfrac {1}{3}})^{1/2}} θ 3 [ exp ( − 2 π ) ] = θ 3 [ q ( 2 − 1 ) ] = 2 π − 1 K ( 2 − 1 ) = 2 − 1 / 8 cos ( 1 8 π ) π − 1 / 2 β ( 3 8 ) 1 / 2 {\displaystyle \theta _{3}[\exp(-{\sqrt {2}}\,\pi )]=\theta _{3}[q({\sqrt {2}}-1)]={\sqrt {2\pi ^{-1}K({\sqrt {2}}-1)}}=2^{-1/8}\cos({\tfrac {1}{8}}\pi )\,\pi ^{-1/2}\beta ({\tfrac {3}{8}})^{1/2}} θ 4 [ exp ( − 2 π ) ] = θ 4 [ q ( 2 − 1 ) ] = 2 2 − 2 4 2 π − 1 K ( 2 − 1 ) = 2 − 1 / 4 cos ( 1 8 π ) 1 / 2 π − 1 / 2 β ( 3 8 ) 1 / 2 {\displaystyle \theta _{4}[\exp(-{\sqrt {2}}\,\pi )]=\theta _{4}[q({\sqrt {2}}-1)]={\sqrt[{4}]{2{\sqrt {2}}-2}}\,{\sqrt {2\pi ^{-1}K({\sqrt {2}}-1)}}=2^{-1/4}\cos({\tfrac {1}{8}}\pi )^{1/2}\,\pi ^{-1/2}\beta ({\tfrac {3}{8}})^{1/2}} Partition sequences and Pochhammer products [ edit ] Regular partition number sequence [ edit ] The regular partition sequenceP ( n ) {\displaystyle P(n)} integer number n {\displaystyle n} n = 1 {\displaystyle n=1} n = 5 {\displaystyle n=5} P {\displaystyle P}
Example values of P(n) and associated number partitions n P(n) paying partitions 0 1 () empty partition/empty sum 1 1 (1) 2 2 (1+1), (2) 3 3 (1+1+1), (1+2), (3) 4 5 (1+1+1+1), (1+1+2), (2+2), (1+3), (4) 5 7 (1+1+1+1+1), (1+1+1+2), (1+2+2), (1+1+3), (2+3), (1+4), (5)
The generating function of the regular partition number sequence can be represented via Pochhammer product in the following way:
∑ k = 0 ∞ P ( k ) x k = 1 ( x ; x ) ∞ = θ 3 ( x ) − 1 / 6 θ 4 ( x ) − 2 / 3 [ θ 3 ( x ) 4 − θ 4 ( x ) 4 16 x ] − 1 / 24 {\displaystyle \sum _{k=0}^{\infty }P(k)x^{k}={\frac {1}{(x;x)_{\infty }}}=\theta _{3}(x)^{-1/6}\theta _{4}(x)^{-2/3}{\biggl [}{\frac {\theta _{3}(x)^{4}-\theta _{4}(x)^{4}}{16\,x}}{\biggr ]}^{-1/24}} The summandization of the now mentionedPochhammer product is described by thePentagonal number theorem in this way:
( x ; x ) ∞ = 1 + ∑ n = 1 ∞ [ − x Fn ( 2 n − 1 ) − x Kr ( 2 n − 1 ) + x Fn ( 2 n ) + x Kr ( 2 n ) ] {\displaystyle (x;x)_{\infty }=1+\sum _{n=1}^{\infty }{\bigl [}-x^{{\text{Fn}}(2n-1)}-x^{{\text{Kr}}(2n-1)}+x^{{\text{Fn}}(2n)}+x^{{\text{Kr}}(2n)}{\bigr ]}} The following basic definitions apply to thepentagonal numbers and the card house numbers:
Fn ( z ) = 1 2 z ( 3 z − 1 ) {\displaystyle {\text{Fn}}(z)={\tfrac {1}{2}}z(3z-1)} Kr ( z ) = 1 2 z ( 3 z + 1 ) {\displaystyle {\text{Kr}}(z)={\tfrac {1}{2}}z(3z+1)} As a further application[ 19]
( x ; x ) 3 = ∏ n = 1 ∞ ( 1 − x n ) 3 = ∑ m = 0 ∞ ( − 1 ) m ( 2 m + 1 ) x m ( m + 1 ) / 2 {\displaystyle (x;x)^{3}=\prod _{n=1}^{\infty }(1-x^{n})^{3}=\sum _{m=0}^{\infty }(-1)^{m}(2m+1)x^{m(m+1)/2}} Strict partition number sequence [ edit ] And the strict partition sequenceQ ( n ) {\displaystyle Q(n)} n {\displaystyle n} [ 20] [ 21]
Example values of Q(n) and associated number partitions n Q(n) Number partitions without repeated summands Number partitions with only odd addends 0 1 () empty partition/empty sum () empty partition/empty sum 1 1 (1) (1) 2 1 (2) (1+1) 3 2 (1+2), (3) (1+1+1), (3) 4 2 (1+3), (4) (1+1+1+1), (1+3) 5 3 (2+3), (1+4), (5) (1+1+1+1+1), (1+1+3), (5) 6 4 (1+2+3), (2+4), (1+5), (6) (1+1+1+1+1+1), (1+1+1+3), (3+3), (1+5) 7 5 (1+2+4), (3+4), (2+5), (1+6), (7) (1+1+1+1+1+1+1), (1+1+1+1+3), (1+3+3), (1+1+5), (7) 8 6 (1+3+4), (1+2+5), (3+5), (2+6), (1+7), (8) (1+1+1+1+1+1+1+1), (1+1+1+1+1+3), (1+1+3+3), (1+1+1+ 5), (3+5), (1+7)
The generating function of the strict partition number sequence can be represented using Pochhammer's product:
∑ k = 0 ∞ Q ( k ) x k = 1 ( x ; x 2 ) ∞ = θ 3 ( x ) 1 / 6 θ 4 ( x ) − 1 / 3 [ θ 3 ( x ) 4 − θ 4 ( x ) 4 16 x ] 1 / 24 {\displaystyle \sum _{k=0}^{\infty }Q(k)x^{k}={\frac {1}{(x;x^{2})_{\infty }}}=\theta _{3}(x)^{1/6}\theta _{4}(x)^{-1/3}{\biggl [}{\frac {\theta _{3}(x)^{4}-\theta _{4}(x)^{4}}{16\,x}}{\biggr ]}^{1/24}} Overpartition number sequence [ edit ] TheMaclaurin series for the reciprocal of the functionϑ01 over partition sequence as coefficients with a positive sign:[ 22]
1 θ 4 ( x ) = ∏ n = 1 ∞ 1 + x n 1 − x n = ∑ k = 0 ∞ P ¯ ( k ) x k {\displaystyle {\frac {1}{\theta _{4}(x)}}=\prod _{n=1}^{\infty }{\frac {1+x^{n}}{1-x^{n}}}=\sum _{k=0}^{\infty }{\overline {P}}(k)x^{k}} 1 θ 4 ( x ) = 1 + 2 x + 4 x 2 + 8 x 3 + 14 x 4 + 24 x 5 + 40 x 6 + 64 x 7 + 100 x 8 + 154 x 9 + 232 x 10 + … {\displaystyle {\frac {1}{\theta _{4}(x)}}=1+2x+4x^{2}+8x^{3}+14x^{4}+24x^{5}+40x^{6}+64x^{7}+100x^{8}+154x^{9}+232x^{10}+\dots } If, for a given numberk {\displaystyle k} [ 23] k {\displaystyle k} P ¯ ( k ) {\displaystyle {\overline {P}}(k)}
First example:
P ¯ ( 4 ) = 14 {\displaystyle {\overline {P}}(4)=14} These 14 possibilities of partition markings exist for the sum 4:
(4), (4 ), (3+1), (3 +1), (3+1 ), (3 +1 ), (2+2), (2 +2), (2+1+1), (2 +1+1), (2+1 +1), (2 +1 +1), (1+1+1+1), (1 +1+1+1)
Second example:
P ¯ ( 5 ) = 24 {\displaystyle {\overline {P}}(5)=24} These 24 possibilities of partition markings exist for the sum 5:
(5), (5 ), (4+1), (4 +1), (4+1 ), (4 +1 ), (3+2), (3 +2), (3+2 ), (3 +2 ), (3+1+1), (3 +1+1), (3+1 +1), (3 +1 +1), (2+2+1), (2 +2+1), (2+2+1 ), (2 +2+1 ),(2+1+1+1), (2 +1+1+1), (2+1 +1+1), (2 +1 +1+1), (1+1+1+1+1), (1 +1+1+1+1)
Relations of the partition number sequences to each other [ edit ] In the Online Encyclopedia of Integer Sequences (OEIS), the sequence of regular partition numbersP ( n ) {\displaystyle P(n)} Q ( n ) {\displaystyle Q(n)} P ¯ ( n ) {\displaystyle {\overline {P}}(n)} n = 1 {\displaystyle n=1}
The sequence of superpartitionsP ¯ ( n ) {\displaystyle {\overline {P}}(n)} [ 24] [ 25]
P ¯ ( n ) = ∑ k = 0 n P ( n − k ) Q ( k ) {\displaystyle {\overline {P}}(n)=\sum _{k=0}^{n}P(n-k)Q(k)} In the following table of sequences of numbers, this formula should be used as an example:
n P(n) Q(n) P ¯ ( n ) {\displaystyle {\overline {P}}(n)} 0 1 1 1 = 1*1 1 1 1 2 = 1 * 1 + 1 * 1 2 2 1 4 = 2 * 1 + 1 * 1 + 1 * 1 3 3 2 8 = 3 * 1 + 2 * 1 + 1 * 1 + 1 * 2 4 5 2 14 = 5 * 1 + 3 * 1 + 2 * 1 + 1 * 2 + 1 * 2 5 7 3 24 = 7 * 1 + 5 * 1 + 3 * 1 + 2 * 2 + 1 * 2 + 1 * 3
Related to this property, the following combination of two series of sums can also be set up via the functionϑ01 :
θ 4 ( x ) = [ ∑ k = 0 ∞ P ( k ) x k ] − 1 [ ∑ k = 0 ∞ Q ( k ) x k ] − 1 {\displaystyle \theta _{4}(x)={\biggl [}\sum _{k=0}^{\infty }P(k)x^{k}{\biggr ]}^{-1}{\biggl [}\sum _{k=0}^{\infty }Q(k)x^{k}{\biggr ]}^{-1}} ^ Tyurin, Andrey N. (30 October 2002). "Quantization, Classical and Quantum Field Theory and Theta-Functions".arXiv :math/0210466v1 ^ Chang, Der-Chen (2011).Heat Kernels for Elliptic and Sub-elliptic Operators . Birkhäuser. p. 7. ^ Tata Lectures on Theta I doi :10.1007/978-0-8176-4577-9 .ISBN 978-0-8176-4572-4 ^ Bagis, Nikolaos D (19 December 2022). "On the Complete Evaluation of Jacobi Theta Functions".arXiv :1503.01141 math.GM ]. ^ Bagis, Nikolaos D (31 July 2019). "Evaluations of Series Related to Jacobi Elliptic Functions".arXiv :1803.09445 math.NT ]. ^ Yi, Jinhee (2004)."Theta-function identities and the explicit formulas for theta-function and their applications" .Journal of Mathematical Analysis and Applications .292 (2):381– 400.doi :10.1016/j.jmaa.2003.12.009 ^ Berndt, Bruce C; Rebák, Örs (9 January 2022)."Explicit Values for Ramanujan's Theta Function ϕ(q)" .Hardy-Ramanujan Journal .44 : 8923.arXiv :2112.11882 doi :10.46298/hrj.2022.8923 S2CID 245851672 . ^ Yi, Jinhee (15 April 2004)."Theta-function identities and the explicit formulas for theta-function and their applications" .Journal of Mathematical Analysis and Applications .292 (2):381– 400.doi :10.1016/j.jmaa.2003.12.009 ^ Andreas Dieckmann:Table of Infinite Products Infinite Sums Infinite Series, Elliptic Theta. ^ Landau (1899) zitiert nachBorwein , Page 94, Exercise 3. ^ "Number-theoretical, combinatorial and integer functions – mpmath 1.1.0 documentation" . Retrieved2021-07-18 .^ Mező, István (2013), "Duplication formulae involving Jacobi theta functions and Gosper'sq -trigonometric functions",Proceedings of the American Mathematical Society 141 (7):2401– 2410,doi :10.1090/s0002-9939-2013-11576-5 ^ Mező, István (2012)."Aq -Raabe formula and an integral of the fourth Jacobi theta function" .Journal of Number Theory .133 (2):692– 704.doi :10.1016/j.jnt.2012.08.025 hdl :2437/166217 ^ Weisstein, Eric W. "Elliptic Alpha Function" .MathWorld ^ "integration - Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$" . 2022-08-13.^ Ohyama, Yousuke (1995)."Differential relations of theta functions" .Osaka Journal of Mathematics .32 (2):431– 450.ISSN 0030-6126 . ^ Shimura, On modular forms of half integral weight ^ "Elliptic Integral Singular Value" .msu.edu . Retrieved2023-04-07 .^ Ramanujan's theta-function identities involving Lambert series ^ "code golf - Strict partitions of a positive integer" . Retrieved2022-03-09 .^ "A000009 - OEIS" . 2022-03-09.^ Mahlburg, Karl (2004). "The overpartition function modulo small powers of 2".Discrete Mathematics .286 (3):263– 267.doi :10.1016/j.disc.2004.03.014 . ^ Kim, Byungchan (28 April 2009)."Elsevier Enhanced Reader" .Discrete Mathematics .309 (8):2528– 2532.doi :10.1016/j.disc.2008.05.007 ^ Eric W. Weisstein (2022-03-11)."Partition Function P" . ^ Eric W. Weisstein (2022-03-11)."Partition Function Q" . Abramowitz, Milton ;Stegun, Irene A. (1964).Handbook of Mathematical Functions Dover Publications .sec. 16.27ff. ISBN 978-0-486-61272-0 {{cite book }}:ISBN / Date incompatibility (help ) Akhiezer, Naum Illyich (1990) [1970].Elements of the Theory of Elliptic Functions .AMS Translations of Mathematical Monographs. Vol. 79. Providence, RI:AMS .ISBN 978-0-8218-4532-5 Farkas, Hershel M. ;Kra, Irwin (1980).Riemann Surfaces . New York:Springer-Verlag . ch. 6.ISBN 978-0-387-90465-8 (for treatment of the Riemann theta) Hardy, G. H. ;Wright, E. M. (1959).An Introduction to the Theory of Numbers (4th ed.). Oxford:Clarendon Press .Mumford, David (1983).Tata Lectures on Theta I . Boston:Birkhauser .ISBN 978-3-7643-3109-2 Pierpont, James (1959).Functions of a Complex Variable . New York:Dover Publications .Rauch, Harry E. ; Farkas, Hershel M. (1974).Theta Functions with Applications to Riemann Surfaces . Baltimore:Williams & Wilkins .ISBN 978-0-683-07196-2 Reinhardt, William P.; Walker, Peter L. (2010),"Theta Functions" , inOlver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248 Whittaker, E. T. ;Watson, G. N. (1927).A Course in Modern Analysis (4th ed.). Cambridge:Cambridge University Press . ch. 21.(history of Jacobi'sθ functions) Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974,ISBN 0-683-07196-3 .
Charles Hermite: Sur la résolution de l'Équation du cinquiéme degré Comptes rendus, C. R. Acad. Sci. Paris, Nr. 11, March 1858. This article incorporates material from Integral representations of Jacobi theta functions onPlanetMath , which is licensed under theCreative Commons Attribution/Share-Alike License .
