In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:

• .. is given in polar form. These logarithms are equally spaced along a vertical line in the complex plane.

• . Such complex logarithm functions are analogous to the real logarithm function, which is the inverse of the real exponential function and hence satisfies for all positive real numbers , or by the process of analytic continuation.

There is no continuous. Ways of dealing with this include branches, the associated Riemann surface, and partial inverses of the complex exponential function. The principal value that is continuous except along the negative real axis; on the complex plane with the negative real numbers and 0 removed, it is the analytic continuation of the (real) natural logarithm.

Problems with inverting the complex exponential function

For a function to have an inverse, it must map distinct values to distinct values; that is, it must be injective and integer radians 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. There are two solutions to this problem.

: this leads naturally to the definition of branches.

Another way to resolve the indeterminacy is to view the logarithm as a function whose domain is not a region in the complex plane, but a Riemann surface that covers 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.

Principal value

Definition

, the principal value is the logarithm whose imaginary part..

is used by some authors

Calculating the principal value

The polar form is the absolute value is its argument. The absolute value is real and positive. The argument is defined up to addition of an integer multiple of . Its principal value is the value that belongs to the interval, which is expressed as \operatorname{atan2}(y,x).

.

The principal value as an inverse function

bijectively. The conformal mapping section below explains the geometric properties of this map in more detail.

The principal value as an analytic continuation

is the analytic 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.

Properties

;

Branches of the complex logarithm

? The answer is no. To see why, imagine tracking such a logarithm function along the unit circle.

of the complex plane. Because one of the goals is to be able to differentiate should be an open set is connected could be unrelated to each other. All this motivates the following definition:

obtained by removing 0 and all negative real numbers from the complex plane.

Another example: The Mercator series converges locally uniformly, as can be shown by differentiating the difference and comparing values at 1.)

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" to have a precise unambiguous meaning.

-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 paper Unwinding the Branches of the Lambert W function and was later referenced in the work of David Jeffrey.

Branch cuts

containing a closed curve that winds has a branch point at 0 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 a branch cut. For example, the principal branch has a branch cut along the negative real axis.

at a negative real number.

Power series for branches

Like every holomorphic function, the complex logarithm can be represented locally – near any point in its domain – with a power series; that is, it is also analytic as the sum of a geometric series., the largest disk not containing the origin (where there is a pole

Antiderivative - constant of power Series

by a power series provides a Taylor series similar to the Mercator series.

Complex logarithm for negative real numbers

.

The derivative of the complex logarithm

. Since the exponential function is holomorphic (that is, complex differentiable) with nonvanishing derivative, the complex analogue of the inverse function theorem. Another way to prove this is to check the Cauchy–Riemann equations in polar coordinates.

Constructing branches via integration

instead.

In developing the analogue for the complex logarithm, there is an additional complication: the definition of the complex integral requires a choice of path. Fortunately, if the integrand is holomorphic, then the value of the integral is unchanged by deforming the path (while holding the endpoints fixed), and in a simply connected (a region with "no holes"), any can be continuously deformed into any other. All this leads to the following:

The complex logarithm as a conformal map

is a conformal map (in the sense that the tangent lines), then the images of the two curves form the same is never 0, it defines a conformal map.

, has the following properties, which are direct consequences of the formula in terms of polar form:

• Circles in the -plane centered at 0 are mapped to vertical segments in the is the real log of the radius of the circle.

• Rays emanating from 0 in the -plane are mapped to horizontal lines in the -plane.

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

The associated Riemann surface

Construction

. These agree on the upper half plane, but not on the lower half plane. So it makes sense to glue the domains of these branches only along the copies of the upper half plane radians counterclockwise around level of the lower half plane.

.

. In this way, .

The logarithm function on the Riemann surface

. by gluing compatible holomorphic functions is known as analytic continuation.

.

Gluing all branches of log z

Instead of gluing only the branches chosen above, one can start with all, and simultaneously glue every as before. This approach, although slightly harder to visualize, is more natural in that it does not require selecting any particular branches.

arises in this way.

The Riemann surface as a universal cover

as a covering space. In fact, it is a Galois covering with deck transformation, generated by the homeomorphism.

As a complex manifold is biholomorphic is the universal cover.

Applications

• The complex logarithm is needed to define exponentiation. has a single value if and only if is an integer.

• Because trigonometric functions can be expressed as rational functions, the inverse trigonometric functions can be expressed in terms of complex logarithms.

• In electrical engineering, the propagation constant involves a complex logarithm.

Generalizations

Logarithms to other bases

Logarithms of holomorphic functions

.

.

Notes

References

• 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.