Inmathematics, acomplex logarithm is a generalization of thenatural logarithm to nonzerocomplex numbers. The term refers to one of the following, which are strongly related:

• A complex logarithm of a nonzero complex number${\displaystyle z}$, defined to be any complex number${\displaystyle w}$ for which${\displaystyle e^w=z}$.^[1][2] Such a number${\displaystyle w}$ is denoted by${\displaystyle \log z}$.^[1] If${\displaystyle z}$ is given inpolar form as${\displaystyle z=r{e}^{i\theta }}$, where${\displaystyle r}$ and${\displaystyle \theta }$ are real numbers with${\displaystyle r>0}$, then${\displaystyle \ln r+i\theta }$ is one logarithm of${\displaystyle z}$, and all the complex logarithms of${\displaystyle z}$ are exactly the numbers of the form${\displaystyle \ln r+i\left(\theta +2\pi k\right)}$ for integers${\displaystyle k}$.^[1][2] These logarithms are equally spaced along a vertical line in the complex plane.
• A complex-valued function${\displaystyle \log :U\to \mathbb{C}}$, defined on some subset${\displaystyle U}$ of the set${\displaystyle {\mathbb{C}}^{\ast }}$ of nonzero complex numbers, satisfying${\displaystyle e^{\log z}=z}$ for all${\displaystyle z}$ in${\displaystyle U}$. Such complex logarithm functions are analogous to the reallogarithm function${\displaystyle \ln :{\mathbb{R}}_{>0}\to \mathbb{R}}$, which is theinverse of the realexponential function and hence satisfiese^lnx =x for all positive real numbersx. Complex logarithm functions can be constructed by explicit formulas involving real-valued functions, by integration of${\displaystyle 1/z}$, or by the process ofanalytic continuation.

There is nocontinuous complex logarithm function defined on all of${\displaystyle {\mathbb{C}}^{\ast }}$. Ways of dealing with this includebranches, the associatedRiemann surface, andpartial inverses of thecomplex exponential function. Theprincipal value defines a particular complex logarithm function${\displaystyle \mathrm{Log}:{\mathbb{C}}^{\ast }\to \mathbb{C}}$ that is continuous except along the negative real axis; on thecomplex plane with the negative real numbers and 0 removed, it is theanalytic continuation of the (real) natural logarithm.

For a function to have an inverse, it mustmap distinct values to distinct values; that is, it must beinjective. But the complex exponential function is not injective, because${\displaystyle e^{w+2\pi ik}=e^w}$ for any complex number${\displaystyle w}$ and integer${\displaystyle k}$, since adding${\displaystyle i\theta }$ to${\displaystyle z}$ has the effect of rotating${\displaystyle e^w}$ counterclockwise${\displaystyle \theta }$radians. So the points

${\displaystyle \dots ,w-4\pi i,w-2\pi i,w,w+2\pi i,w+4\pi i,\dots ,}$

equally spaced along a vertical line, are all mapped to the same number by the exponential function. This means that the exponential function does not have an inverse function in the standard sense.^[3][4] There are two solutions to this problem.

One is to restrict the domain of the exponential function to a region that does not contain any two numbers differing by an integer multiple of${\displaystyle 2\pi \mathit{i}}$: this leads naturally to the definition ofbranches of${\displaystyle \log z}$, which are certain functions that single out one logarithm of each number in their domains. This is analogous to the definition of${\displaystyle \arcsin x}$ on${\displaystyle [-1,1]}$ as the inverse of the restriction of${\displaystyle \sin \theta }$ to the interval${\displaystyle [-\pi /2,\pi /2]}$: there are infinitely many real numbers${\displaystyle \theta }$ with${\displaystyle \sin \theta =x}$, but one arbitrarily chooses the one in${\displaystyle [-\pi /2,\pi /2]}$.

Another way to resolve the indeterminacy is to view the logarithm as a function whose domain is not a region in thecomplex plane, but a Riemann surface thatcovers the punctured complex plane in an infinite-to-1 way.

Branches have the advantage that they can be evaluated at complex numbers. On the other hand, the function on the Riemann surface is elegant in that it packages together all branches of the logarithm and does not require an arbitrary choice as part of its definition.

For each nonzero complex number${\displaystyle z}$, theprincipal value${\displaystyle \mathrm{Log}z}$ is the logarithm whoseimaginary part lies in the interval${\displaystyle (-\pi ,\pi ]}$.^[2] The expression${\displaystyle \mathrm{Log}0}$ is left undefined since there is no complex number${\displaystyle w}$ satisfying${\displaystyle e^w=0}$.^[1]

When the notation${\displaystyle \log z}$ appears without any particular logarithm having been specified, it is generally best to assume that the principal value is intended. In particular, this gives a value consistent with the real value of${\displaystyle \ln z}$ when${\displaystyle z}$ is a positive real number. The capitalization in the notation${\displaystyle \text{Log}}$ is used by some authors^[2] to distinguish the principal value from other logarithms of${\displaystyle z.}$

Thepolar form of a nonzero complex number${\displaystyle z=x+yi}$ is${\displaystyle z=r{e}^{i\theta }}$, where$r=|z|=\sqrt{x^2+y^2}$ is theabsolute value of${\displaystyle z}$, and${\displaystyle \theta }$ is itsargument. The absolute value is real and positive. The argument is definedup to addition of an integer multiple of2π. Itsprincipal value is the value that belongs to theinterval${\displaystyle (-\pi ,\pi ]}$, which is expressed as${\displaystyle \mathrm{atan2}(y,x)}$.

This leads to the following formula for the principal value of the complex logarithm:

${\displaystyle \mathrm{Log}z=\ln r+i\theta =\ln |z|+i\mathrm{Arg}z=\ln \sqrt{x^2+y^2}+i\mathrm{atan2}(y,x).}$

For example,${\displaystyle \mathrm{Log}(-3i)=\ln 3-\pi i/2}$, and${\displaystyle \mathrm{Log}(-3)=\ln 3+\pi i}$.

Another way to describe${\displaystyle \mathrm{Log}z}$ is as the inverse of a restriction of the complex exponential function, as in the previous section. The horizontal strip${\displaystyle S}$ consisting of complex numbers${\displaystyle w=x+yi}$ such that is an example of a region not containing any two numbers differing by an integer multiple of${\displaystyle 2\pi i}$, so the restriction of the exponential function to${\displaystyle S}$ has an inverse. In fact, the exponential function maps${\displaystyle S}$bijectively to the punctured complex plane${\displaystyle {\mathbb{C}}^{\ast }=\mathbb{C}\setminus \{0\}}$, and the inverse of this restriction is${\displaystyle \mathrm{Log}:{\mathbb{C}}^{\ast }\to S}$. The conformal mapping section below explains the geometric properties of this map in more detail.

On the region${\displaystyle \mathbb{C}-{\mathbb{R}}_{\le 0}}$ consisting of complex numbers that are not negative real numbers or 0, the function${\displaystyle \mathrm{Log}z}$ is theanalytic continuation of the natural logarithm. The values on the negative real line can be obtained as limits of values at nearby complex numbers with positive imaginary parts.

Not all identities satisfied by${\displaystyle \ln }$ extend to complex numbers. It is true that${\displaystyle e^{\mathrm{Log}z}=z}$ for all${\displaystyle z\ne 0}$ (this is what it means for${\displaystyle \mathrm{Log}z}$ to be a logarithm of${\displaystyle z}$), but the identity${\displaystyle \mathrm{Log}(e^z)=z}$ fails for${\displaystyle z}$ outside the strip${\displaystyle S}$. For this reason, one cannot always apply${\displaystyle \text{Log}}$ to both sides of an identity${\displaystyle e^z=e^w}$ to deduce${\displaystyle z=w}$. Also, the identity${\displaystyle \mathrm{Log}(z_1{z}_2)=\mathrm{Log}{z}_1+\mathrm{Log}{z}_2}$ can fail: the two sides can differ by an integer multiple of${\displaystyle 2\pi i}$;^[1] for instance,

${\displaystyle \mathrm{Log}((-1)i)=\mathrm{Log}(-i)=\ln (1)-\frac{\pi i}{2}=-\frac{\pi i}{2},}$

but

${\displaystyle \mathrm{Log}(-1)+\mathrm{Log}(i)=\left(\ln (1)+\pi i\right)+\left(\ln (1)+\frac{\pi i}{2}\right)=\frac{3\pi i}{2}\ne -\frac{\pi i}{2}.}$

The function${\displaystyle \mathrm{Log}z}$ is discontinuous at each negative real number, but continuous everywhere else in${\displaystyle {\mathbb{C}}^{\ast }}$. To explain the discontinuity, consider what happens to${\displaystyle \arg z}$ as${\displaystyle z}$ approaches a negative real number${\displaystyle a}$. If${\displaystyle z}$ approaches${\displaystyle a}$ from above, then${\displaystyle \arg z}$ approaches${\displaystyle \pi ,}$ which is also the value of${\displaystyle \arg a}$ itself. But if${\displaystyle z}$ approaches${\displaystyle a}$ from below, then${\displaystyle \arg z}$ approaches${\displaystyle -\pi .}$ So${\displaystyle \arg z}$ "jumps" by${\displaystyle 2\pi }$ as${\displaystyle z}$ crosses the negative real axis, and similarly${\displaystyle \mathrm{Log}z}$ jumps by${\displaystyle 2\pi i.}$

Is there a different way to choose a logarithm of each nonzero complex number so as to make a function${\displaystyle \mathrm{L}(z)}$ that is continuous on all of${\displaystyle {\mathbb{C}}^{\ast }}$? The answer is no. To see why, imagine tracking such a logarithm function along theunit circle, by evaluating${\displaystyle \mathrm{L}\left(e^{i\theta }\right)}$ as${\displaystyle \theta }$ increases from${\displaystyle 0}$ to${\displaystyle 2\pi }$. If${\displaystyle \mathrm{L}(z)}$ is continuous, then so is${\displaystyle \mathrm{L}\left(e^{i\theta }\right)-i\theta }$, but the latter is a difference of two logarithms of${\displaystyle e^{i\theta },}$ so it takes values in the discrete set${\displaystyle 2\pi i\mathbb{Z},}$ so it is constant. In particular,${\displaystyle \mathrm{L}\left(e^{2\pi i}\right)-2\pi i=\mathrm{L}\left(e^0\right)-0}$, which contradicts${\displaystyle \mathrm{L}\left(e^{2\pi i}\right)=\mathrm{L}\left(e^0\right)}$.

To obtain a continuous logarithm defined on complex numbers, it is hence necessary to restrict the domain to a smaller subset${\displaystyle U}$ of the complex plane. Because one of the goals is to be able todifferentiate the function, it is reasonable to assume that the function is defined on a neighborhood of each point of its domain; in other words,${\displaystyle U}$ should be anopen set. Also, it is reasonable to assume that${\displaystyle U}$ isconnected, since otherwise the function values on different components of${\displaystyle U}$ could be unrelated to each other. All this motivates the following definition:

Abranch of${\displaystyle \log z}$ is a continuous function${\displaystyle \mathrm{L}(z)}$ defined on a connected open subset${\displaystyle U}$ of the complex plane such that${\displaystyle \mathrm{L}(z)}$ is a logarithm of${\displaystyle z}$ for each${\displaystyle z}$ in${\displaystyle U}$.^[2]

For example, the principal value defines a branch on the open set where it is continuous, which is the set${\displaystyle \mathbb{C}-{\mathbb{R}}_{\le 0}}$ obtained by removing 0 and all negative real numbers from the complex plane.

Another example: TheMercator series

${\displaystyle \ln (1+u)=\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n}{u}^n=u-\frac{u^2}{2}+\frac{u^3}{3}-\cdots }$

convergeslocally uniformly for, so setting${\displaystyle z=1+u}$ defines a branch of${\displaystyle \log z}$ on the open disk of radius 1 centered at 1. (Actually, this is just a restriction of${\displaystyle \mathrm{Log}z}$, as can be shown by differentiating the difference and comparing values at 1.)

Once a branch is fixed, it may be denoted${\displaystyle \log z}$ if no confusion can result. Different branches can give different values for the logarithm of a particular complex number, however, so a branch must be fixed in advance (or else the principal branch must be understood) in order for "${\displaystyle \log z}$" to have a precise unambiguous meaning.

In some literature, the notation${\displaystyle {\ln }_{k}z}$ is used to explicitly denote the${\displaystyle k}$-th branch of the complex logarithm. This notation is particularly useful when working with multi-valued logarithms in complex analysis and topology. It was first introduced in the paperUnwinding the Branches of the Lambert W function^[5] and was later referenced in the work ofDavid Jeffrey.^[6]

The argument above involving the unit circle generalizes to show that no branch of${\displaystyle \log z}$ exists on an open set${\displaystyle U}$ containing aclosed curve thatwinds around 0. One says that${\displaystyle \log z}$ has abranch point at 0. To avoid containing closed curves winding around 0,${\displaystyle U}$ is typically chosen as the complement of a ray or curve in the complex plane going from 0 (inclusive) to infinity in some direction. In this case, the curve is known as abranch cut. For example, the principal branch has a branch cut along the negative real axis.

If the function${\displaystyle \mathrm{L}(z)}$ is extended to be defined at a point of the branch cut, it will necessarily be discontinuous there; at best it will be continuous "on one side", like${\displaystyle \mathrm{Log}z}$ at a negative real number.

Each branch${\displaystyle \mathrm{L}(z)}$ of${\displaystyle \log z}$ on an open set${\displaystyle U}$ is the inverse of a restriction of the exponential function, namely the restriction to the image${\displaystyle \mathrm{L}(U)}$. Since the exponential function isholomorphic (that is, complex differentiable) with nonvanishing derivative, the complex analogue of theinverse function theorem applies. It shows that${\displaystyle \mathrm{L}(z)}$ is holomorphic on${\displaystyle U}$, and${\displaystyle {\mathrm{L}}^{\prime }(z)=1/z}$ for each${\displaystyle z}$ in${\displaystyle U}$.^[2] Another way to prove this is to check theCauchy–Riemann equations in polar coordinates.^[2]

The function${\displaystyle \ln (x)}$ for real${\displaystyle x>0}$ can be constructed by the formula$${\displaystyle \ln (x)={\int }_1^x\frac{du}{u}.}$$If the range of integration started at a positive number${\displaystyle a}$ other than 1, the formula would have to be$${\displaystyle \ln (x)=\ln (a)+{\int }_a^x\frac{du}{u}}$$instead.

In developing the analogue for thecomplex logarithm, there is an additional complication: the definition of thecomplex integral requires a choice of path. Fortunately, if the integrand is holomorphic, then the value of the integral is unchanged bydeforming the path (while holding the endpoints fixed), and in asimply connected region${\displaystyle U}$ (a region with "no holes"),any path from${\displaystyle a}$ to${\displaystyle z}$ inside${\displaystyle U}$ can becontinuously deformed inside${\displaystyle U}$ into any other. All this leads to the following:

If${\displaystyle U}$ is a simply connected open subset of${\displaystyle \mathbb{C}}$ not containing 0, then a branch of${\displaystyle \log z}$ defined on${\displaystyle U}$ can be constructed by choosing a starting point${\displaystyle a}$ in${\displaystyle U}$, choosing a logarithm${\displaystyle b}$ of${\displaystyle a}$, and defining$${\displaystyle L(z)=b+{\int }_a^z\frac{dw}{w}}$$ for each${\displaystyle z}$ in${\displaystyle U}$.^[7]

Any holomorphic map${\displaystyle f:U\to \mathbb{C}}$ satisfying${\displaystyle f^{\prime }(z)\ne 0}$ for all${\displaystyle z\in U}$ is aconformal map, which means that if two curves passing through a point${\displaystyle a}$ of${\displaystyle U}$ form an angle${\displaystyle \alpha }$ (in the sense that thetangent lines to the curves at${\displaystyle a}$ form an angle${\displaystyle \alpha }$), then the images of the two curves form thesame angle${\displaystyle \alpha }$ at${\displaystyle f(a)}$.Since a branch of${\displaystyle \log z}$ is holomorphic, and since its derivative${\displaystyle 1/z}$ is never 0, it defines a conformal map.

For example, the principal branch${\displaystyle w=\mathrm{Log}z}$, viewed as a mapping from${\displaystyle \mathbb{C}-{\mathbb{R}}_{\le 0}}$ to the horizontal strip defined by, has the following properties, which are direct consequences of the formula in terms of polar form:

• Circles^[8] in thez-plane centered at 0 are mapped to vertical segments in thew-plane connecting${\displaystyle a-\pi i}$ to${\displaystyle a+\pi i}$, where${\displaystyle a}$ is the real log of the radius of the circle.
• Rays emanating from 0 in thez-plane are mapped to horizontal lines in thew-plane.

Each circle and ray in thez-plane as above meet at a right angle. Their images under Log are a vertical segment and a horizontal line (respectively) in thew-plane, and these too meet at a right angle. This is an illustration of the conformal property of Log.

The various branches of${\displaystyle \log z}$ cannot be glued to give a single continuous function${\displaystyle \log :{\mathbb{C}}^{\ast }\to \mathbb{C}}$ because two branches may give different values at a point where both are defined. Compare, for example, the principal branch${\displaystyle \mathrm{Log}z}$ on${\displaystyle \mathbb{C}-{\mathbb{R}}_{\le 0}}$ with imaginary part${\displaystyle \theta }$ in${\displaystyle (-\pi ,\pi )}$ and the branch${\displaystyle \mathrm{L}(z)}$ on${\displaystyle \mathbb{C}-{\mathbb{R}}_{\ge 0}}$ whose imaginary part${\displaystyle \theta }$ lies in${\displaystyle (0,2\pi )}$. These agree on theupper half plane, but not on the lower half plane. So it makes sense to glue the domains of these branchesonly along the copies of the upper half plane. The resulting glued domain is connected, but it has two copies of the lower half plane. Those two copies can be visualized as two levels of a parking garage, and one can get from the${\displaystyle \text{Log}}$ level of the lower half plane up to the${\displaystyle \text{L}}$ level of the lower half plane by going${\displaystyle 2\pi }$ radians counterclockwise around0, first crossing the positive real axis (of the${\displaystyle \text{Log}}$ level) into the shared copy of the upper half plane and then crossing the negative real axis (of the${\displaystyle \text{L}}$ level) into the${\displaystyle \text{L}}$ level of the lower half plane.

One can continue by gluing branches with imaginary part${\displaystyle \theta }$ in${\displaystyle (\pi ,3\pi )}$, in${\displaystyle (2\pi ,4\pi )}$, and so on, and in the other direction, branches with imaginary part${\displaystyle \theta }$ in${\displaystyle (-2\pi ,0)}$, in${\displaystyle (-3\pi ,-\pi )}$, and so on. The final result is a connected surface that can be viewed as a spiraling parking garage with infinitely many levels extending both upward and downward. This is the Riemann surface${\displaystyle R}$ associated to${\displaystyle \log z}$.^[9]

A point on${\displaystyle R}$ can be thought of as a pair${\displaystyle (z,\theta )}$ where${\displaystyle \theta }$ is a possible value of the argument of${\displaystyle z}$. In this way,R can be embedded in${\displaystyle \mathbb{C}\times \mathbb{R}\approx {\mathbb{R}}^3}$.

Because the domains of the branches were glued only along open sets where their values agreed, the branches glue to give a single well-defined function${\displaystyle {\log }_R:R\to \mathbb{C}}$.^[10] It maps each point${\displaystyle (z,\theta )}$ on${\displaystyle R}$ to${\displaystyle \ln |z|+i\theta }$. This process of extending the original branch${\displaystyle \text{Log}}$ by gluing compatible holomorphic functions is known asanalytic continuation.

There is a "projection map" from${\displaystyle R}$ down to${\displaystyle {\mathbb{C}}^{\ast }}$ that "flattens" the spiral, sending${\displaystyle (z,\theta )}$ to${\displaystyle z}$. For any${\displaystyle z\in {\mathbb{C}}^{\ast }}$, if one takes all the points${\displaystyle (z,\theta )}$ of${\displaystyle R}$ lying "directly above"${\displaystyle z}$ and evaluates${\displaystyle {\log }_R}$ at all these points, one gets all the logarithms of${\displaystyle z}$.

Instead of gluing only the branches chosen above, one can start withall branches of${\displaystyle \log z}$, and simultaneously glueevery pair of branches${\displaystyle L_1:U_1\to \mathbb{C}}$ and${\displaystyle L_2:U_2\to \mathbb{C}}$ along the largest open subset of${\displaystyle U_1\cap U_2}$ on which${\displaystyle L_1}$ and${\displaystyle L_2}$ agree. This yields the same Riemann surface${\displaystyle R}$ and function${\displaystyle {\log }_R}$ as before. This approach, although slightly harder to visualize, is more natural in that it does not require selecting any particular branches.

If${\displaystyle U^{\prime }}$ is an open subset of${\displaystyle R}$ projecting bijectively to its image${\displaystyle U}$ in${\displaystyle {\mathbb{C}}^{\ast }}$, then the restriction of${\displaystyle {\log }_R}$ to${\displaystyle U^{\prime }}$ corresponds to a branch of${\displaystyle \log z}$ defined on${\displaystyle U}$. Every branch of${\displaystyle \log z}$ arises in this way.

The projection map${\displaystyle R\to {\mathbb{C}}^{\ast }}$ realizes${\displaystyle R}$ as acovering space of${\displaystyle {\mathbb{C}}^{\ast }}$. In fact, it is aGalois covering withdeck transformation group isomorphic to${\displaystyle \mathbb{Z}}$, generated by thehomeomorphism sending${\displaystyle (z,\theta )}$ to${\displaystyle (z,\theta +2\pi )}$.

As acomplex manifold,${\displaystyle R}$ isbiholomorphic with${\displaystyle \mathbb{C}}$ via${\displaystyle {\log }_R}$. (The inverse map sends${\displaystyle z}$ to${\displaystyle \left(e^z,\mathrm{Im}(z)\right)}$.) This shows that${\displaystyle R}$ is simply connected, so${\displaystyle R}$ is theuniversal cover of${\displaystyle {\mathbb{C}}^{\ast }}$.

• The complex logarithm is needed to defineexponentiation in which the base is a complex number. Namely, if${\displaystyle a}$ and${\displaystyle b}$ are complex numbers with${\displaystyle a\ne 0}$, one can use the principal value to define${\displaystyle a^b=e^{b\mathrm{Log}a}}$. One can also replace${\displaystyle \mathrm{Log}a}$ by other logarithms of${\displaystyle a}$ to obtain other values of${\displaystyle a^b}$, differing by factors of the form${\displaystyle e^{2\pi inb}}$.^[1][11] The expression${\displaystyle a^b}$ has a single value if and only if${\displaystyle b}$ is an integer.^[1]
• Becausetrigonometric functions can be expressed asrational functions of${\displaystyle e^{iz}}$, theinverse trigonometric functions can be expressed in terms of complex logarithms.
• In electrical engineering, thepropagation constant involves a complex logarithm.

Just as for real numbers, one can define for complex numbers${\displaystyle b}$ and${\displaystyle x}$

${\displaystyle {\log }_{b}x=\frac{\log x}{\log b},}$

with the only caveat that its value depends on the choice of a branch of log defined at${\displaystyle b}$ and${\displaystyle x}$ (with${\displaystyle \log b\ne 0}$). For example, using the principal value gives

${\displaystyle {\log }_{i}e=\frac{\mathrm{Log}e}{\mathrm{Log}i}=\frac{1}{\pi i/2}=-\frac{2i}{\pi }.}$

Iff is a holomorphic function on a connected open subset${\displaystyle U}$ of${\displaystyle \mathbb{C}}$, then a branch of${\displaystyle \log f}$ on${\displaystyle U}$ is a continuous function${\displaystyle g}$ on${\displaystyle U}$ such that${\displaystyle e^{g(z)}=f(z)}$ for all${\displaystyle z}$ in${\displaystyle U}$. Such a function${\displaystyle g}$ is necessarily holomorphic with${\displaystyle g^{\prime }(z)=f^{\prime }(z)/f(z)}$ for all${\displaystyle z}$ in${\displaystyle U}$.

If${\displaystyle U}$ is a simply connected open subset of${\displaystyle \mathbb{C}}$, and${\displaystyle f}$ is a nowhere-vanishing holomorphic function on${\displaystyle U}$, then a branch of${\displaystyle \log f}$ defined on${\displaystyle U}$ can be constructed by choosing a starting pointa in${\displaystyle U}$, choosing a logarithm${\displaystyle b}$ of${\displaystyle f(a)}$, and defining

${\displaystyle g(z)=b+{\int }_a^z\frac{f^{\prime }(w)}{f(w)}dw}$

for each${\displaystyle z}$ in${\displaystyle U}$.^[2]

• Calkin, Neil J.;Chan, Eunice Y. S.;Corless, Robert M. (2023).Computational Discovery on Jupyter. Society for Industrial and Applied Mathematics. ISBN 978-1-61197-749-3.
• Jeffrey, D. J.;Hare, D. E. G.;Corless, Robert M. (1996).Unwinding the branches of the Lambert W function. The Mathematical Scientist,21, 1–7.
• Ahlfors, Lars V. (1966).Complex Analysis (2nd ed.). McGraw-Hill.
• Conway, John B. (1978).Functions of One Complex Variable (2nd ed.). Springer.ISBN 9780387903286.
• Kreyszig, Erwin (2011).Advanced Engineering Mathematics (10th ed.). Berlin:Wiley.ISBN 9780470458365.
• Lang, Serge (1993).Complex Analysis (3rd ed.). Springer-Verlag.ISBN 9783642592737.
• Moretti, Gino (1964).Functions of a Complex Variable. Prentice-Hall.
• Sarason, Donald (2007).Complex Function Theory (2nd ed.). American Mathematical Society.ISBN 9780821886229.
• Whittaker, E. T.;Watson, G. N. (1927).A Course of Modern Analysis (Fourth ed.). Cambridge University Press.

• Analytic functions
• Logarithms

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