Real part of log((1-i)/(1+i)) is (a) 1 (b) 0 (c) (1)/(2) (d) (1)/(4)

Name: Problem 21, Chapter 1: IIT JEE Super Course in Mathematics ALGEBRA II
Uploaded: 2021-06-04T16:39:40-08:00
Duration: 5 min 9 s
Channel: Anas Venkitta
Description: Real part of log((1-i)/(1+i)) is (a) 1 (b) 0 (c) (1)/(2) (d) (1)/(4)

step 1 So in the given question we are told that the logarithm of a complex number, the logarithm of a

Real part of log((1-i)/(1+i)) is (a) 1 (b) 0 (c) (1)/(2) (d)...

Key Concepts

Properties of Quotients in the Complex Plane

When dividing two complex numbers, the moduli divide and the arguments subtract. This property is crucial in simplifying the logarithm of a quotient of complex numbers. It allows one to extract the real part by taking the logarithm of the quotient of the moduli, thus focusing on the computation of ln(r1/r2) without handling the angular components unless necessary.

Polar Representation of Complex Numbers

Representing complex numbers in polar form involves expressing them in terms of their modulus (absolute value) and argument (angle). This approach simplifies operations such as multiplication or division, as the modulus and arguments follow separate arithmetic rules (moduli multiply/divide and arguments add/subtract), which is particularly useful when computing expressions like the quotient of two complex numbers.

Complex Logarithm

The logarithm of a complex number is defined through its polar form representation, where any non-zero complex number can be written as z = r e^(i?). The logarithm then becomes log z = ln r + i?, where the real part is the natural logarithm of the modulus (r) and the imaginary part is the argument (?). This concept is central when dealing with logarithms in the complex plane.

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