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 2πi 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 elogx = x. One writes w = logz 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 2πi.

That value of w = logz 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 = logz with Im w ∈ [0, 2π) called the main value.

A positive real number x has the logarithms logx + 2kπi, k ∈ & integers; while a negative real number x the logarithms log |x| + (2k + 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) = logx + logy, x, y > 0 caution is advised in the complex. are z1, z2 ∈ & Copf; *, then log (z1z2) and logz1 + logz2 generally by an additive integer multiple of 2πi. For Re z1 > 0, Re z2 But> 0 applies