Problem 4 - Advanced Engineering Mathematics

Question

The analytic function \(f(z)=\cosh z\) is conformal except at \(z=\)

Step-by-Step Solution

Answer

The function is not conformal at points where the derivative is zero, i.e., at integer multiples of \( \pi i \).

Step 1: Understanding Conformality

A function is conformal at a point if it is analytic and its derivative is non-zero at that point. The function \( f(z) = \cosh z \) is known to be analytic, so we need to examine where its derivative is zero.

Step 2: Differentiating the Function

Find the derivative of \( f(z) = \cosh z \). The derivative is \( f'(z) = \sinh z \), since the derivative of \( \cosh z \) is \( \sinh z \).

Step 3: Solving for Zero Derivative

Set \( f'(z) = \sinh z = 0 \) and solve for \( z \). The hyperbolic sine function is zero at integer multiples of \( \pi i \), i.e., \( z = n \pi i \) where \( n \) is an integer.

Key Concepts

Analytic Function

An analytic function is a type of complex function that is smooth and differentiable throughout its domain. This holds true not only for a single point but also within an entire neighborhood around that point. Because of this characteristic, these functions lend themselves beautifully to complex analysis.
An important property of analytic functions is that they can be represented as power series around any point within their domain. Consider the function of interest, \( f(z) = \cosh z \).
Since it is composed of exponential functions, \( \cosh z \) is inherently analytic everywhere in the complex plane.

• Being analytic means \( \cosh z \) is infinitely differentiable.
• Its analytic nature ensures that conformal mappings—and any associated consequences, like the preservation of angles—apply except where the derivative becomes zero.

Remember, determining the zeros of its derivative, \( \sinh z \), is crucial to understanding where the function may fail to be conformal.

Derivative of Hyperbolic Functions

Hyperbolic functions, which have direct parallels to trigonometric functions, arise often in complex analysis and have properties similar to their circular counterparts. The function \( \cosh z \) is the hyperbolic equivalent of \( \cos(z) \), and differentiating it gives rise to another hyperbolic function.
The derivative of \( \cosh z \) is \( \sinh z \), based on the following identity:

• \[ \frac{d}{dz}(\cosh z) = \sinh z \].

This derivative isn't just a mathematical curiosity; it’s a conduit to understanding when and where \( \cosh z \) ceases to be conformal — specifically when \( \sinh z \) equals zero.Hyperbolic derivatives also underlie many physical phenomena and geometric interpretations due to their symmetry and utility in describing hyperbolas.

Zeros of Hyperbolic Sine

The zeros of a hyperbolic sine function, \( \sinh z \), are those values of \( z \) that make the function equal to zero. Solving \( \sinh z = 0 \) involves understanding the mathematical identity of hyperbolic sine:

• \[ \sinh z = \frac{e^z - e^{-z}}{2} \].

The result \( \sinh z = 0 \) occurs whenever the quantity \( e^z = e^{-z} \), leading to solutions that are integer multiples of imaginary \( \pi \):

• \( z = n\pi i \), where \( n \) is an integer.

What’s remarkable is that these zeros highlight the precise points where the analytic function \( \cosh z \) fails to be conformal due to a zero derivative, as conformality requires the derivative to be non-zero. This specific insight is crucial when evaluating the behavior of complex functions in different domains.