The value of log(-i) is (a) -(π)/(2) (b) (π)/(2) (c) -i (π)/(2) (d) i (π)/(2)

Name: Problem 15, Chapter 1: IIT JEE Super Course in Mathematics ALGEBRA II
Uploaded: 2021-06-06T14:44:07-08:00
Duration: 3 min 13 s
Channel: Anas Venkitta
Description: The value of log(-i) is (a) -(π)/(2) (b) (π)/(2) (c) -i (π)/(2) (d) i (π)/(2)

step 1 So in the given question we are told to find the value of the expression log of minus i. So let'

The value of log(-i) is (a) -(π)/(2) (b) (π)/(2) (c) -i...

Key Concepts

Complex Logarithm

The complex logarithm extends the real logarithm to complex numbers. It is defined by expressing a complex number in polar form and then writing its logarithm as the sum of the logarithm of the modulus and an imaginary part involving the argument. This definition makes the logarithm a multi-valued function, which is crucial in complex analysis.

Polar Representation of Complex Numbers

Any non-zero complex number can be expressed in the form r·exp(i?), where r is the modulus (absolute value) and ? is the argument (angle). This representation is key in computing functions like the logarithm and the exponential in the complex plane.

Principal Value

The principal value of a multi-valued function, such as the complex logarithm, is a conventionally chosen branch that provides a unique value. In the context of the logarithm, it typically involves restricting the argument to lie within a specified interval, ensuring consistency across calculations.

Multi-valued Functions and Branch Cuts

Functions in complex analysis, like the logarithm, can take on multiple values due to the periodic nature of the complex exponential function. To handle this, branch cuts are defined to restrict the domain in a way that makes these functions single-valued over that domain. This concept is fundamental for understanding the behavior and computation of complex logarithms.

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