Table Of Content
Brush Up Basics
Let a + ib be a complex number whose logarithm is to be found.
Step 1: Convert the given complex number, into polar form.
Where, Amplitude is
and argument is
Step 2: Use Euler’s Theorem to rewrite complex number in polar form to exponential form.
There r (cos θ + i sin θ) is written as reiθ. This means that
a+ ib = reiθ
Step 3: Take logarithm of both sides we get.
The above results can be expressed in terms of modulus and argument of z.
Examples to clear it all
Find the logarithm of 1 + i?
Step 1: Convert 1 + i into polar form
Step 2: Use Euler’s Theorem to rewrite complex number
Step 3: Take logarithm of both sides
Find the logarithm of iota, i
Step 1: Convert i into polar form
Step 2: Rewrite iota into exponential form
Step 3: Take logarithm of both sides.
Observations to give you insight
Why should we convert a complex number to its exponential form? The answer is simple: Exponentials are also known as anti-logarithms. It requires no brilliance that taking log of anti-log gives us the log of that number.
The logarithm of a complex number can be a real number only if
Argument of a complex number can only be zero if its imaginary part, b is zero.