Complex logarithm (base e)

When z is written in polar coordinates, one can easily derive its logarithm :

z = |z| e = √(x2 +y2) e

u = lnz = ln √(x2 +y2) + i(ϕ + k 2π); periodic in i k 2π , k integer

Φ = atan (y/x)

Principal value for k = 0: u = lnz = ln √(x2 +y2) + iϕ

Because of the periodicity of e with i2kπ, the logarithm has an infinite number of branches, which differ in k. The branch with k = 0 is called the principal value. This is the default situation when the simulation is opened.

Integer k can be freely chosen in the number field k . The simulation will then show the k branch of the logarithm. The initialize button resets all parameters to their default values except k.

With real numbers the logarithm exists for positive numbers only. This is demonstrated by contracting the quadratic array to a point, and shifting it along the real axis. With positive values > 1 in the z plane one recognizes the logarithmic shrinkage along the positive real axis in the u plane. With x <1 the logarithm becomes negative and diverges at x = 0 (use the number field and input x = 0.000001, then x = 0 ). With x < 0 the logarithm becomes a complex number with imaginary value y = iπ. With x  = - 1 it is purely imaginary: ln (-1) = iπ. In the complex plane the logarithm exists for any number.

After initialization the lowest row of the quadratic array is at the real axis with x > 0. All points of this row are mapped to positive or negative points of the real axis in the u-plane. Shifting the array along the x axis in positive direction, one recognizes the shrinking along both axis. Use proper scales and inputs to the x, y number fields to see this for larger values of the logarithm. Shifting in the negative x direction a point at the real axis jumps to the line y = iπ beyond the divergence x = 0 and moves in a positive direction.

All in all one recognizes that the area of the z plane with x > 1 is mapped to a horizontal strip in the u-plane with positive real part, symmetric to the real axis, and limited by he curve x = 0.5 ln(1 + y2), y = arctan y, which is the mapping of points with x = 1. This limit curve is shown green when the check box visible is activated . At the same time a second curve is shown in yellow, which is the mapping of points with equal imaginary value y = 1 in the z-plane; in the u-plane x = 0.5 ln (1 + x2), y = atan (1/x).

The area of the plane with x < -1 is mapped to two partial areas of the positive u-plane, that are limited by limit curves shifted by ∓ π in y direction.

The area inside of the unit circle is mapped to the negative real half plane, within the strip ∓ iπ .

A sufficiently complex picture, to stimulate the ambition for deeper investigation and understanding! Study of the circular array is interesting, too!.

Using different k values confirms the periodicity of the logarithm. The z-plane is mapped to the k strip in the u-plane.