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Holomorphic function

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Complex-differentiable (mathematical) function

For Zariski's theory of holomorphic functions on an algebraic variety, seeformal holomorphic function.

"Holomorphism" redirects here and is not to be confused withHomomorphism.

A rectangular grid (top) and its image under aconformal map⁠ $f {\displaystyle f}$ ⁠ (bottom).

A rectangular grid (top) and its image under aconformal map⁠ $f {\displaystyle f}$ ⁠ (bottom).

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Mapping of the function $f(z)={\frac {1}{z}}$ . The animation shows different $z {\displaystyle z}$ in blue color with the corresponding $f(z)$ in red color. The point $z {\displaystyle z}$ and $f(z)$ are shown in the $\mathbb {C} {\tilde {=}}\mathbb {R} ^{2}$ . y-axis represents the imaginary part of the complex number of $z {\displaystyle z}$ and $f(z)$ .

{\displaystyle f(z)={\frac {1}{z}}} — Mapping of the function $f(z)={\frac {1}{z}}$ . The animation shows different $z {\displaystyle z}$ in blue color with the corresponding $f(z)$ in red color. The point $z {\displaystyle z}$ and $f(z)$ are shown in the $\mathbb {C} {\tilde {=}}\mathbb {R} ^{2}$ . y-axis represents the imaginary part of the complex number of $z {\displaystyle z}$ and $f(z)$ .

Inmathematics, aholomorphic function is acomplex-valued function of one ormore complex variables that iscomplex differentiable in aneighbourhood of each point in adomain incomplex coordinate space⁠ $\mathbb {C} ^{n}$ ⁠. The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function isinfinitely differentiable and locally equal to its ownTaylor series (isanalytic). Holomorphic functions are the central objects of study incomplex analysis.

Though the termanalytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergentpower series in a neighbourhood of each point in itsdomain. That all holomorphic functions are complex analytic functions, and vice versa, is amajor theorem in complex analysis.^[1]

Holomorphic functions are also sometimes referred to asregular functions.^[2] A holomorphic function whose domain is the wholecomplex plane is called anentire function. The phrase "holomorphic at a point⁠ $z_{0}$ ⁠" means not just differentiable at⁠ $z_{0}$ ⁠, but differentiable everywhere within some close neighbourhood of⁠ $z_{0}$ ⁠ in the complex plane.

Definition

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The function⁠ $f(z)={\bar {z}}$ ⁠ is not complex differentiable at zero, because as shown above, the value of⁠ ${\frac {f(z)-f(0)}{z-0}}$ ⁠ varies depending on the direction from which zero is approached. On the real axis only,⁠ $f {\displaystyle f}$ ⁠ equals the function⁠ $g(z)=z$ ⁠ and the limit is⁠ $1 {\displaystyle 1}$ ⁠, while along the imaginary axis only,⁠ $f {\displaystyle f}$ ⁠ equals the different function⁠ $h(z)=-z$ ⁠ and the limit is⁠ $-1$ ⁠. Other directions yield yet other limits.

{\displaystyle f(z)={\bar {z}}} — The function⁠ $f(z)={\bar {z}}$ ⁠ is not complex differentiable at zero, because as shown above, the value of⁠ ${\frac {f(z)-f(0)}{z-0}}$ ⁠ varies depending on the direction from which zero is approached. On the real axis only,⁠ $f {\displaystyle f}$ ⁠ equals the function⁠ $g(z)=z$ ⁠ and the limit is⁠ $1 {\displaystyle 1}$ ⁠, while along the imaginary axis only,⁠ $f {\displaystyle f}$ ⁠ equals the different function⁠ $h(z)=-z$ ⁠ and the limit is⁠ $-1$ ⁠. Other directions yield yet other limits.

Given a complex-valued function⁠ $f {\displaystyle f}$ ⁠ of a single complex variable, thederivative of⁠ $f {\displaystyle f}$ ⁠ at a point⁠ $z_{0}$ ⁠ in its domain is defined as thelimit^[3]

f'(z_{0})=\lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}.

This is the same definition as for thederivative of areal function, except that all quantities are complex. In particular, the limit is taken as the complex number⁠ $z {\displaystyle z}$ ⁠ tends to⁠ $z_{0}$ ⁠, and this means that the same value is obtained for any sequence of complex values for⁠ $z {\displaystyle z}$ ⁠ that tends to⁠ $z_{0}$ ⁠. If the limit exists,⁠ $f {\displaystyle f}$ ⁠ is said to becomplex differentiable at⁠ $z_{0}$ ⁠. This concept of complex differentiability shares several properties withreal differentiability: It islinear and obeys theproduct rule,quotient rule, andchain rule.^[4]

A function isholomorphic on anopen set⁠ $U {\displaystyle U}$ ⁠ if it iscomplex differentiable atevery point of⁠ $U {\displaystyle U}$ ⁠. A function⁠ $f {\displaystyle f}$ ⁠ isholomorphic at a point⁠ $z_{0}$ ⁠ if it is holomorphic on someneighbourhood of⁠ $z_{0}$ ⁠.^[5]A function isholomorphic on some non-open set⁠ $A {\displaystyle A}$ ⁠ if it is holomorphic at every point of⁠ $A {\displaystyle A}$ ⁠.

A function may be complex differentiable at a point but not holomorphic at this point. For example, the function $\textstyle f(z)=|z|{\vphantom {l}}^{2}=z{\bar {z}}$ is complex differentiable at⁠ $0 {\displaystyle 0}$ ⁠, butis not complex differentiable anywhere else, esp. including in no place close to⁠ $0 {\displaystyle 0}$ ⁠ (see the Cauchy–Riemann equations, below). So, it isnot holomorphic at⁠ $0 {\displaystyle 0}$ ⁠.

The relationship between real differentiability and complex differentiability is the following: If a complex function⁠ $f(x+iy)=u(x,y)+i\,v(x,y)$ ⁠ is holomorphic, then⁠ $u {\displaystyle u}$ ⁠ and⁠ $v {\displaystyle v}$ ⁠ have first partial derivatives with respect to⁠ $x {\displaystyle x}$ ⁠ and⁠ $y {\displaystyle y}$ ⁠, and satisfy theCauchy–Riemann equations:^[6]

{\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}\qquad {\mbox{and}}\qquad {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}\,

or, equivalently, theWirtinger derivative of⁠ $f {\displaystyle f}$ ⁠ with respect to⁠ ${\bar {z}}$ ⁠, thecomplex conjugate of⁠ $z {\displaystyle z}$ ⁠, is zero:^[7]

{\frac {\partial f}{\partial {\bar {z}}}}=0,

which is to say that, roughly,⁠ $f {\displaystyle f}$ ⁠ is functionally independent from⁠ ${\bar {z}}$ ⁠, the complex conjugate of⁠ $z {\displaystyle z}$ ⁠.

If continuity is not given, the converse is not necessarily true. A simple converse is that if⁠ $u {\displaystyle u}$ ⁠ and⁠ $v {\displaystyle v}$ ⁠ havecontinuous first partial derivatives and satisfy the Cauchy–Riemann equations, then⁠ $f {\displaystyle f}$ ⁠ is holomorphic. A more satisfying converse, which is much harder to prove, is theLooman–Menchoff theorem: if⁠ $f {\displaystyle f}$ ⁠ is continuous,⁠ $u {\displaystyle u}$ ⁠ and⁠ $v {\displaystyle v}$ ⁠ have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then⁠ $f {\displaystyle f}$ ⁠ is holomorphic.^[8]

Terminology

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The termholomorphic was introduced in 1875 byCharles Briot andJean-Claude Bouquet, two ofAugustin-Louis Cauchy's students, and derives from the Greekὅλος (hólos) meaning "whole", andμορφή (morphḗ) meaning "form" or "appearance" or "type", in contrast to the termmeromorphic derived fromμέρος (méros) meaning "part". A holomorphic function resembles anentire function ("whole") in adomain of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolatedpoles), resembles a rational fraction ("part") of entire functions in a domain of the complex plane.^[9] Cauchy had instead used the termsynectic.^[10]

Today, the term "holomorphic function" is sometimes preferred to "analytic function". An important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow obviously from the definitions. The term "analytic" is however also in wide use.

Properties

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Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.^[11] That is, if functions⁠ $f {\displaystyle f}$ ⁠ and⁠ $g {\displaystyle g}$ ⁠ are holomorphic in a domain⁠ $U {\displaystyle U}$ ⁠, then so are⁠ $f+g$ ⁠,⁠ $f-g$ ⁠,⁠ $f g {\displaystyle fg}$ ⁠, and⁠ $f\circ g$ ⁠. Furthermore,⁠ $f/g$ ⁠ is holomorphic if⁠ $g {\displaystyle g}$ ⁠ has no zeros in⁠ $U {\displaystyle U}$ ⁠; otherwise it ismeromorphic.

If one identifies⁠ $\mathbb {C}$ ⁠ with the realplane⁠ $\textstyle \mathbb {R} ^{2}$ ⁠, then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve theCauchy–Riemann equations, a set of twopartial differential equations.^[6]

Every holomorphic function can be separated into its real and imaginary parts⁠ $f(x+iy)=u(x,y)+i\,v(x,y)$ ⁠, and each of these is aharmonic function on⁠ $\textstyle \mathbb {R} ^{2}$ ⁠ (each satisfiesLaplace's equation⁠ $\textstyle \nabla ^{2}u=\nabla ^{2}v=0$ ⁠), with⁠ $v {\displaystyle v}$ ⁠ theharmonic conjugate of⁠ $u {\displaystyle u}$ ⁠.^[12]Conversely, every harmonic function⁠ $u(x,y)$ ⁠ on asimply connected domain⁠ $\textstyle \Omega \subset \mathbb {R} ^{2}$ ⁠ is the real part of a holomorphic function: If⁠ $v {\displaystyle v}$ ⁠ is the harmonic conjugate of⁠ $u {\displaystyle u}$ ⁠, unique up to a constant, then⁠ $f(x+iy)=u(x,y)+i\,v(x,y)$ ⁠ is holomorphic.

Cauchy's integral theorem implies that thecontour integral of every holomorphic function along aloop vanishes:^[13]

\oint _{\gamma }f(z)\,\mathrm {d} z=0.

Here⁠ $\gamma$ ⁠ is arectifiable path in a simply connectedcomplex domain⁠ $U\subset \mathbb {C}$ ⁠ whose start point is equal to its end point, and⁠ $f\colon U\to \mathbb {C}$ ⁠ is a holomorphic function.

Cauchy's integral formula states that every function holomorphic inside adisk is completely determined by its values on the disk's boundary.^[13] Furthermore: Suppose⁠ $U\subset \mathbb {C}$ ⁠ is a complex domain,⁠ $f\colon U\to \mathbb {C}$ ⁠ is a holomorphic function and the closed disk $D\equiv \{z:|z-z_{0}|\leq r\}$ iscompletely contained in⁠ $U {\displaystyle U}$ ⁠. Let⁠ $\gamma$ ⁠ be the circle forming theboundary of⁠ $D {\displaystyle D}$ ⁠. Then for every⁠ $a {\displaystyle a}$ ⁠ in theinterior of⁠ $D {\displaystyle D}$ ⁠:

f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,\mathrm {d} z

where the contour integral is takencounter-clockwise.

The derivative⁠ ${f'}(a)$ ⁠ can be written as a contour integral^[13] usingCauchy's differentiation formula:

f'\!(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(z-a)^{2}}}\,\mathrm {d} z,

for any simple loop positively winding once around⁠ $a {\displaystyle a}$ ⁠, and

f'\!(a)=\lim \limits _{\gamma \to a}{\frac {i}{2{\mathcal {A}}(\gamma )}}\oint _{\gamma }f(z)\,\mathrm {d} {\bar {z}},

forinfinitesimal positive loops⁠ $\gamma$ ⁠ around⁠ $a {\displaystyle a}$ ⁠.

In regions where the first derivative is not zero, holomorphic functions areconformal: they preserve angles and the shape (but not size) of small figures.^[14]

Everyholomorphic function is analytic. That is, a holomorphic function⁠ $f {\displaystyle f}$ ⁠ has derivatives of every order at each point⁠ $a {\displaystyle a}$ ⁠ in its domain, and it coincides with its ownTaylor series at⁠ $a {\displaystyle a}$ ⁠ in a neighbourhood of⁠ $a {\displaystyle a}$ ⁠. In fact,⁠ $f {\displaystyle f}$ ⁠ coincides with its Taylor series at⁠ $a {\displaystyle a}$ ⁠ in any disk centred at that point and lying within the domain of the function.

From an algebraic point of view, the set of holomorphic functions on an open set is acommutative ring and acomplex vector space. Additionally, the set of holomorphic functions in an open set⁠ $U {\displaystyle U}$ ⁠ is anintegral domain if and only if the open set⁠ $U {\displaystyle U}$ ⁠ is connected.^[7] In fact, it is alocally convex topological vector space, with theseminorms being thesuprema oncompact subsets.

From a geometric perspective, a function⁠ $f {\displaystyle f}$ ⁠ is holomorphic at⁠ $z_{0}$ ⁠ if and only if itsexterior derivative⁠ $\mathrm {d} f$ ⁠ in a neighbourhood⁠ $U {\displaystyle U}$ ⁠ of⁠ $z_{0}$ ⁠ is equal to⁠ $f'(z)\,\mathrm {d} z$ ⁠ for some continuous function⁠ $f^{'} {\displaystyle f'}$ ⁠. It follows from

0=\mathrm {d} ^{2}f=\mathrm {d} (f'\,\mathrm {d} z)=\mathrm {d} f'\wedge \mathrm {d} z

that⁠ $\mathrm {d} f'$ ⁠ is also proportional to⁠ $\mathrm {d} z$ ⁠, implying that the derivative⁠ $\mathrm {d} f'$ ⁠ is itself holomorphic and thus that⁠ $f {\displaystyle f}$ ⁠ is infinitely differentiable. Similarly,⁠ $\mathrm {d} (f\,\mathrm {d} z)=f'\,\mathrm {d} z\wedge \mathrm {d} z=0$ ⁠ implies that any function⁠ $f {\displaystyle f}$ ⁠ that is holomorphic on the simply connected region⁠ $U {\displaystyle U}$ ⁠ is also integrable on⁠ $U {\displaystyle U}$ ⁠.

(For a path⁠ $\gamma$ ⁠ from⁠ $z_{0}$ ⁠ to⁠ $z {\displaystyle z}$ ⁠ lying entirely in⁠ $U {\displaystyle U}$ ⁠, define⁠ $F_{\gamma }(z)=F(0)+\int _{\gamma }f\,\mathrm {d} z$ ⁠; in light of theJordan curve theorem and thegeneralized Stokes' theorem,⁠ $F_{\gamma }(z)$ ⁠ is independent of the particular choice of path⁠ $\gamma$ ⁠, and thus⁠ $F(z)$ ⁠ is a well-defined function on⁠ $U {\displaystyle U}$ ⁠ having⁠ $\mathrm {d} F=f\,\mathrm {d} z$ ⁠ or⁠ $f={\frac {\mathrm {d} F}{\mathrm {d} z}}$ ⁠.)

Examples

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Allpolynomial functions in⁠ $z {\displaystyle z}$ ⁠ with complexcoefficients areentire functions (holomorphic in the whole complex plane⁠ $\mathbb {C}$ ⁠), and so are theexponential function⁠ $\exp z$ ⁠ and thetrigonometric functions⁠ $\cos {z}={\tfrac {1}{2}}{\bigl (}\exp(+iz)+\exp(-iz){\bigr )}$ ⁠ and⁠ $\sin {z}=-{\tfrac {1}{2}}i{\bigl (}\exp(+iz)-\exp(-iz){\bigr )}$ ⁠ (cf.Euler's formula). Theprincipal branch of thecomplex logarithm function⁠ $\log z$ ⁠ is holomorphic on the domain⁠ $\mathbb {C} \smallsetminus \{z\in \mathbb {R} :z\leq 0\}$ ⁠. Thesquare root function can be defined as⁠ ${\sqrt {z}}\equiv \exp {\bigl (}{\tfrac {1}{2}}\log z{\bigr )}$ ⁠ and is therefore holomorphic wherever the logarithm⁠ $\log z$ ⁠ is. Thereciprocal function⁠ ${\tfrac {1}{z}}$ ⁠ is holomorphic on⁠ $\mathbb {C} \smallsetminus \{0\}$ ⁠. (The reciprocal function, and any otherrational function, ismeromorphic on⁠ $\mathbb {C}$ ⁠.)

As a consequence of theCauchy–Riemann equations, any real-valued holomorphic function must beconstant. Therefore, theabsolute value $|z|$ , theargument⁠ $\arg z$ ⁠, thereal part⁠ $\operatorname {Re} (z)$ ⁠ and theimaginary part⁠ $\operatorname {Im} (z)$ ⁠ are not holomorphic. Another typical example of acontinuous function which is not holomorphic is thecomplex conjugate⁠ ${\bar {z}}.$ ⁠ (The complex conjugate isantiholomorphic.)

Several variables

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The definition of a holomorphic function generalizes to several complex variables in a straightforward way. A function⁠ $f\colon (z_{1},z_{2},\ldots ,z_{n})\mapsto f(z_{1},z_{2},\ldots ,z_{n})$ ⁠ in⁠ $n {\displaystyle n}$ ⁠ complex variables isanalytic at a point⁠ $p {\displaystyle p}$ ⁠ if there exists a neighbourhood of⁠ $p {\displaystyle p}$ ⁠ in which⁠ $f {\displaystyle f}$ ⁠ is equal to a convergent power series in⁠ $n {\displaystyle n}$ ⁠ complex variables;^[15]the function⁠ $f {\displaystyle f}$ ⁠ isholomorphic in an open subset⁠ $U {\displaystyle U}$ ⁠ of⁠ $\mathbb {C} ^{n}$ ⁠ if it is analytic at each point in⁠ $U {\displaystyle U}$ ⁠.Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function⁠ $f {\displaystyle f}$ ⁠, this is equivalent to⁠ $f {\displaystyle f}$ ⁠ being holomorphic in each variable separately (meaning that if any⁠ $n-1$ ⁠ coordinates are fixed, then the restriction of⁠ $f {\displaystyle f}$ ⁠ is a holomorphic function of the remaining coordinate). The much deeperHartogs' theorem proves that the continuity assumption is unnecessary:⁠ $f {\displaystyle f}$ ⁠ is holomorphic if and only if it is holomorphic in each variable separately.

More generally, a function of several complex variables that issquare integrable over everycompact subset of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions.

Functions of several complex variables are in some basic ways more complicated than functions of a single complex variable. For example, the region of convergence of a power series is not necessarily an open ball; these regions are logarithmically-convexReinhardt domains, the simplest example of which is apolydisk. However, they also come with some fundamental restrictions. Unlike functions of a single complex variable, the possible domains on which there are holomorphic functions that cannot be extended to larger domains are highly limited. Such a set is called adomain of holomorphy.

Acomplex differential⁠ $(p,0)$ ⁠-form⁠ $\alpha$ ⁠ is holomorphic if and only if its antiholomorphicDolbeault derivative is zero:⁠ ${\bar {\partial }}\alpha =0$ ⁠.

Extension to functional analysis

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Main article:infinite-dimensional holomorphy

The concept of a holomorphic function can be extended to the infinite-dimensional spaces offunctional analysis. For instance, theFréchet orGateaux derivative can be used to define a notion of a holomorphic function on aBanach space over the field of complex numbers.

References

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^"Analytic functions of one complex variable".Encyclopedia of Mathematics. European Mathematical Society / Springer. 2015 – via encyclopediaofmath.org.
^"Analytic function",Encyclopedia of Mathematics,EMS Press, 2001 [1994], retrievedFebruary 26, 2021
^Ahlfors, L.,Complex Analysis, 3 ed. (McGraw-Hill, 1979).
^Henrici, P. (1986) [1974, 1977].Applied and Computational Complex Analysis. Wiley. Three volumes, publ.: 1974, 1977, 1986.
^Ebenfelt, Peter; Hungerbühler, Norbert; Kohn, Joseph J.; Mok, Ngaiming; Straube, Emil J. (2011).Complex Analysis. Science & Business Media. Springer.ISBN 978-3-0346-0009-5 – via Google.
^^a ^bMarkushevich, A.I. (1965).Theory of Functions of a Complex Variable. Prentice-Hall. [In three volumes.]
^^a ^bGunning, Robert C.; Rossi, Hugo (1965).Analytic Functions of Several Complex Variables. Modern Analysis. Englewood Cliffs, NJ:Prentice-Hall.ISBN 9780821869536.MR 0180696.Zbl 0141.08601 – via Google.
^Gray, J.D.; Morris, S.A. (April 1978). "When is a function that satisfies the Cauchy-Riemann equations analytic?".The American Mathematical Monthly.85 (4):246–256.doi:10.2307/2321164.JSTOR 2321164.
^The original French terms wereholomorphe andméromorphe.
Briot, Charles Auguste;Bouquet, Jean-Claude (1875)."§15 fonctions holomorphes".Théorie des fonctions elliptiques (2nd ed.). Gauthier-Villars. pp. 14–15.Lorsqu'une fonction est continue, monotrope, et a une dérivée, quand la variable se meut dans une certaine partie du plan, nous dirons qu'elle estholomorphe dans cette partie du plan. Nous indiquons par cette dénomination qu'elle est semblable aux fonctions entières qui jouissent de ces propriétés dans toute l'étendue du plan. [...] ¶ Une fraction rationnelle admet comme pôles les racines du dénominateur; c'est une fonction holomorphe dans toute partie du plan qui ne contient aucun de ses pôles. ¶ Lorsqu'une fonction est holomorphe dans une partie du plan, excepté en certains pôles, nous dirons qu'elle estméromorphe dans cette partie du plan, c'est-à-dire semblable aux fractions rationnelles. [When a function is continuous,monotropic, and has a derivative, when the variable moves in a certain part of the [complex] plane, we say that it isholomorphic in that part of the plane. We mean by this name that it resemblesentire functions which enjoy these properties in the full extent of the plane. [...] ¶ A rational fraction admits aspoles theroots of the denominator; it is a holomorphic function in all that part of the plane which does not contain any poles. ¶ When a function is holomorphic in part of the plane, except at certain poles, we say that it ismeromorphic in that part of the plane, that is to say it resembles rational fractions.]
Harkness, James;Morley, Frank (1893)."5. Integration".A Treatise on the Theory of Functions. Macmillan. p. 161.
^Briot & Bouquet had previously also adopted Cauchy’s termsynectic (synectique in French), in the 1859 first edition of their book.
Briot, Charles Auguste;Bouquet, Jean-Claude (1859)."§10".Théorie des fonctions doublement périodiques. Mallet-Bachelier. p. 11.
^Henrici, Peter (1993) [1986].Applied and Computational Complex Analysis. Wiley Classics Library. Vol. 3 (Reprint ed.). New York - Chichester - Brisbane - Toronto - Singapore:John Wiley & Sons.ISBN 0-471-58986-1.MR 0822470.Zbl 1107.30300 – via Google.
^Evans, L.C. (1998).Partial Differential Equations. American Mathematical Society.
^^a ^b ^cLang, Serge (2003).Complex Analysis. Springer Verlag GTM.Springer Verlag.
^Rudin, Walter (1987).Real and Complex Analysis (3rd ed.). New York: McGraw–Hill Book Co.ISBN 978-0-07-054234-1.MR 0924157.
^Gunning and Rossi.Analytic Functions of Several Complex Variables. p. 2.