Last modified on March 23rd, 2024

chapter outline

 

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) = ln(r) + i(θ + 2nℼ)

Here,

z = r ⋅ e = 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) = ln(r) + i(θ + 2nℼ)

Here,

z = r ⋅ e = 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 = ew …..(i)

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

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

Thus, from equation (i), we get 

eu + iv = r ⋅ e

⇒ eu ⋅ eiv = r ⋅ e

On comparing both sides, we get

eu = r and eiv = e

⇒ 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 = ew

⇒ 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’.

Solve the given complex logarithmic function log(z) for ${z=2\sqrt{3}+2i}$

Solution:

Here, ${z=2\sqrt{3}+2i}$, then ${r=\sqrt{16}}$ and θ = ${\tan ^{-1}\left( \dfrac{2}{2\sqrt{3}}\right)}$ = ${\dfrac{\pi }{6}}$
Thus, ${\log \left( 2\sqrt{3}+2i\right) =\ln \sqrt{16}+i\left( \dfrac{\pi }{6}+2n\pi \right)}$
⇒ ${\log \left( 2\sqrt{3}+2i\right) =\ln \sqrt{16}+\left( \dfrac{1}{6}+2n\right) \pi i}$, where n Є ℤ

Calculate the value of log(1)

Solution:

As we know, e0 = 1
Thus, log(1) = 0
Also, we know e2ℼi = 1
⇒ log(1) = 2ℼi, another possible answer.
Thus, log(1) = 2nℼi, where n Є ℤ

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 = ew 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.

Solve: Log(-1)

Solution:

Here, z = -1 = -1 + 0i, then r = 1 and θ = ${\tan ^{-1}\left( \dfrac{0}{-1}\right)}$ = ℼ
Now, Log(-1) = ln(1) + iθ = 0 + ℼi = ℼi
Thus, Log(-1) = ℼi

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 elog(z) = z ⇒ ew log(z) = zw, where ‘w’ is another complex number function …..(vi)

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

Let us calculate elog(z) and log(ez) for z = 1 + 5i

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

Find the value of zw, for the complex numbers z = 1 + i and w = i

Solution:

As we know, zw = ew log(z)
⇒ (1 + i)i = ei log(1 + i)
Here, ${r=\sqrt{2}}$ and θ = ${\tan ^{-1}\left( \dfrac{1}{1}\right)}$ = ${\dfrac{\pi }{4}}$ 
log(1 + i) = ${\ln \left( \sqrt{2}\right) +i\left( \dfrac{\pi }{4}+2n\pi \right)}$
Now, ei log(1 + i)
= ${e^{i\left( \ln \left( \sqrt{2}\right) +i\left( \dfrac{\pi }{4}+2n\pi \right) \right) }}$
= ${e^{\left( -\dfrac{\pi }{4}+i\left( \ln \sqrt{2}\right) +2n\pi \right) }}$
= ${e^{-\dfrac{\pi }{4}}\cdot e^{i\ln \sqrt{2}}\cdot e^{2n\pi }}$
= ${e^{-\dfrac{\pi }{4}}\cdot \left( \sqrt{2}\right) ^{i}\cdot e^{2n\pi }}$, for n Є ℤ.

Properties

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

Product Rulelog(z1 ⋅ z2) = log(z1) + log(z2)
Quotient Rule${\log \left( \dfrac{z_{1}}{z_{2}}\right) =\log z_{1}-\log z_{2}}$
Power Rulelog(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.

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