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Last modified on April 30th, 2024

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)**

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’.

**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, e^{0} = 1

Thus, log(1) = 0

Also, we know e^{2ℼi} = 1

⇒ log(1) = 2ℼi, another possible answer.

Thus, log(1) = 2nℼi, where n Є ℤ

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.

**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

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).

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.

**Find the value of z ^{w}, for the complex numbers z = 1 + i and w = i**

Solution:

As we know, z^{w} = e^{w log(z)}

⇒ (1 + i)^{i} = e^{i 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, e^{i 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.

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

Product Rule | log(z_{1} ⋅ z_{2}) = log(z_{1}) + log(z_{2}) |

Quotient Rule | ${\log \left( \dfrac{z_{1}}{z_{2}}\right) =\log z_{1}-\log z_{2}}$ |

Power Rule | log(z^{n}) = n ⋅ log(z), where n Є ℤ |

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.

Last modified on April 30th, 2024