# What is the use of complex logarithm

## Logarithm of a complex number

to a complex number *z* ≠ 0 is a number *w* ∈ & Copf; such that

Since the exponential function maps the complex plane ∈ surjectively to the dotted plane & Copf; * = & Copf; \ {0} and the period 2*πi* owns exist to everyone *z* ∈ & Copf; * always an infinite number of logarithms *w* ∈ & Copf ;. Any such *w* is of the shape

Where argz is an argument of *z*, and for *x* > 0 is log *x* the uniquely determined real number with *e*^{log}* ^{x}* =

*x*. One writes

*w*= log

*z*taking into account the ambiguity. (Note that in a function-theoretical context the (natural) logarithm (logarithm function) is usually denoted by log, while in the tradition of real analysis it is usually denoted by ln). Two logarithms each of

*z*differ by an additive integer multiple of 2

*πi*.

That value of *w* = log*z* with Im *w* ∈ (−*π*, *π*] is called the principal value of the logarithm of *z*. The term log is also used for this *z*. Occasionally the value of *w* = log*z* with Im *w* ∈ [0, 2*π*) called the main value.

A positive real number *x* has the logarithms log*x* + 2*kπi*, *k* ∈ & integers; while a negative real number *x* the logarithms log |*x*| + (2*k* + 1)*πi*, *k* ∈ & integers; owns. Furthermore \ (\ text {Log} \, i = \ frac {\ pi} {2} i \) applies.

When applying the law of logarithms known from the real log (*xy*) = log*x* + log*y*, *x*, *y* > 0 caution is advised in the complex. are *z*_{1}, *z*_{2} ∈ & Copf; *, then log (*z*_{1}*z*_{2}) and log*z*_{1} + log*z*_{2} generally by an additive integer multiple of 2*πi*. For Re *z*_{1} > 0, Re *z*_{2} But> 0 applies

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