Complex Logarithm

The complex logarithm is an extension of the concept of logarithmic functions involving complex numbers (represented by log z).

Mathematically, written as

log(z) = log(r ⋅ e^iθ) = ln(r) + i(θ + 2nℼ)

Here,

z = r ⋅ e^iθ = the complex number

r = |z| = the absolute value of z

θ = arg(z) = the argument of z and -ℼ < θ ≤ ℼ

ln(r) = real part

i(θ + 2nℼ) = imaginary part

n = multiple branches of the complex logarithm and n Є ℤ 

log(z) = log(r ⋅ e^iθ) = ln(r) + i(θ + 2nℼ)

Here,

z = r ⋅ e^iθ = the complex number

r = |z| = the absolute value of z

θ = arg(z) = the argument of z and -ℼ < θ ≤ ℼ

ln(r) = real part

i(θ + 2nℼ) = imaginary part

n = multiple branches of the complex logarithm and n Є ℤ

Thus, it is also written as log(z) = ln|z| + i(arg (z) + 2nℼ)

Derivation

Let us consider a non-zero complex number ‘z,’ such that z = e^w …..(i)

If θ = arg(z) with -ℼ < θ ≤ ℼ, then ‘z’ and ‘w’ are expressed as

z = r ⋅ e^iθ and w = u + iv, where ‘u’ is the real part, and ‘iv’ is the imaginary part

Thus, from equation (i), we get 

e^{u + iv} = r ⋅ e^iθ

⇒ e^u ⋅ e^iv = r ⋅ e^iθ

On comparing both sides, we get

e^u = r and e^iv = e^iθ

⇒ u = ln(r) and v = θ + 2nℼ, where n Є ℤ.

It follows that the equation (i) is satisfied if and only if w = ln(r) + i(θ + 2nℼ), where n Є ℤ.

Thus, the equation (i) becomes,

z = e^w

⇒ log(z) = w

⇒ log(z) = ln(r) + i(θ + 2nℼ), where n Є ℤ …..(ii)

⇒ log(z) = ln|z| + i(arg (z) + 2nπ) …..(iii)

Thus, the logarithm of a non-zero complex number z in its general form is derived,

It is thus an example of a multiple-valued function, where all the multiple values of the complex logarithm have the same real part ln(r) but differ in the imaginary part by ‘2ℼ’.

Unique Features

Principal Branch

When n = 0, the principal branch (or the principal value) of log(z), represented by Log(z), is obtained from the equation log(z) = ln(r) + i(θ + 2nℼ) as:

Log(z) = ln(r) + iθ …..(iv)

Thus, the equation z = e^w can also be written as log(z) = Log(z) + 2nℼi, where n Є ℤ …..(v)

Here, the principal branch of the log (Log(z)) is evaluated from the principal branch of the arg, where -ℼ < θ ≤ ℼ.

Branch Cuts

For log (z), infinity is the branch point. Since the function log(z) is infinity only at z = 0, the loop must enclose the origin. 

Here, the branch cuts along the negative real axis, implying a discontinuity along this axis due to the multivalued nature of log(z).

Complex Powers

Let ‘z’ be a non-zero complex number function such that e^log(z) = z ⇒ e^{w log(z)} = z^w, where ‘w’ is another complex number function …..(vi)

Also, log(e^z) = z + 2nℼi, where n Є ℤ …..(vii)

Let us calculate e^log(z) and log(e^z) for z = 1 + 5i

Here, e^log(z) = z = 1 + 5i and log(e^z) = z + 2nℼi = 1 + 5i + 2nℼi = 1 + i(5 + 2nℼ), for n Є ℤ.

Properties

Like the real-valued logarithmic functions, the complex logarithms hold the same properties as discussed below.

Product Rule	log(z1 ⋅ z2) = log(z1) + log(z2)
Quotient Rule	${\log \left( \dfrac{z_{1}}{z_{2}}\right) =\log z_{1}-\log z_{2}}$
Power Rule	log(zn) = n ⋅ log(z), where n Є ℤ

Mappings

A mapping of a function shows how the elements of the domain are connected with the elements of the range. If w = log(z), the domain is z, and the range is w. 

Here, some examples of the mapping of log(z) are:

1) Here, we observe that a point ‘z’ is mapped to many values of ‘w,’ and the dots are represented as:

log(1) → blue dots, log(8) → red dots, log(i) → blue cross, and log(8i) → red cross. 

The values in the principal branch are inside the shaded region in the w-plane, and the values of log(z) for a given ‘z’ are placed at intervals of ‘2πi’ in the w-plane.

2) Here, the circles centered on 0 are mapped on the y-axis, and rays from the origin are mapped on the x-axis. 

We observe that the circles are mapped to the y-axis in the principal branch, and rays are mapped to the x-axis in the principal region of the w-plane, which is shaded as shown.