Complex logarithm

The complex logarithm of a nonzero complex number is the multivalued set of values \(\ln|z|+i(\arg z+2\pi k)\), with integer \(k\). Its branches arise from the periodicity of the complex exponential.

Complex exponential: The function \(e^z\) on complex numbers, periodic with imaginary period \(2\pi i\). Its repeated values are exactly why each nonzero input has infinitely many logarithms.
Polar form: A representation of a complex number by its magnitude and angle. The logarithm separates naturally into the logarithm of magnitude and the angle.
Complex powers: Powers with complex exponents, commonly defined using the complex logarithm and exponential. Their multiple possible values come from the logarithm used in the exponent.
Real logarithm: The inverse of the real exponential, defined for positive real inputs. Unlike the complex logarithm, it has one real value for each positive input.
Argument of a complex number: The angle a nonzero complex number makes with the positive real axis, determined up to multiples of \(2\pi\). The logarithm’s imaginary parts are the possible arguments of its input.
Complex plane: The plane whose points represent complex numbers, with real and imaginary coordinates. Angles and distances in this plane determine the logarithm’s real and imaginary parts.
Contour integration: Integration of complex-valued functions along oriented curves in the complex plane. Logarithmic branches affect whether an integrand can remain analytic along a contour.
Principal logarithm: The single-valued complex logarithm using the principal argument, usually chosen in \((-\pi,\pi]\). It is one conventional branch, not the entire set of complex logarithm values.
Branch of the logarithm: A single-valued choice of logarithm made on a domain where the choice varies analytically. A branch selects one value from the multivalued logarithm at each point in a suitable domain.
Euler's formula: The identity \(e^{ix}=\cos x+i\sin x\) relating complex exponentials to trigonometric functions. It shows that changing an angle by \(2\pi\) leaves the exponential unchanged.

Linked from 17 pages