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Chapter 17: Problem 17

Find all points at which the mapping \(w=\cosh \pi z\) is not conformal.

Short Answer

Expert verified
The mapping is not conformal at points \(z = ni\) where \(n\) is an integer.

Step by step solution

01

Understanding Conformality

A mapping is conformal at a point if it's analytic and its derivative is non-zero at that point. For the function \(w = \, \cosh \, \pi z\), we need to check where these conditions are not satisfied.
02

Check Analyticity

The hyperbolic cosine function is defined as \( \cosh \pi z = \frac{e^{\pi z} + e^{-\pi z}}{2} \). This function is entire, meaning it is analytic everywhere in the complex plane. Therefore, we can say that \(w = \, \cosh \, \pi z\) is analytic at all points in the complex plane.
03

Find the Derivative

To find where the function is not conformal, calculate the derivative: \(\frac{dw}{dz} = \pi \sinh \pi z\). We need to find where this derivative is equal to zero, as conformality fails where the derivative is zero.
04

Solve for the Zero Derivative

Set the derivative to zero: \(\pi \sinh \pi z = 0\). This implies \(\sinh \pi z = 0\). The solutions to this equation are \(\pi z = n\pi i\) for integer \(n\), which simplifies to \(z = ni\), where \(n\) is an integer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Analytic Function
An analytic function is a type of complex function that is differentiable at every point in its domain, not just at isolated points.
This differentiability must also hold true in some neighborhood around each point.
  • An important characteristic of analytic functions is that they can be represented as a power series.
  • Because of this, they are also called holomorphic functions.
In our problem, the function is given as \( w = \cosh \pi z \), which is expressed using hyperbolic cosine, \( \cosh z = \frac{e^z + e^{-z}}{2} \).
This function is entire, meaning it is analytic across the entire complex plane.
Thus, there are no points in the complex plane where the function fails to be analytic.
Complex Plane
The complex plane is a fundamental tool in complex analysis that allows for the visualization and analysis of complex numbers.
Each complex number \( z = x + yi \) can be seen as a point or a vector on this plane, where \( x \) is the real part and \( y \) is the imaginary part.
  • The horizontal axis (real axis) represents the real component.
  • The vertical axis (imaginary axis) represents the imaginary component.
When working with functions like \( w = \cosh \pi z \), we analyze how they transform points from one region of the complex plane to another.
Since this function is entire, it is continually defined and differentiable throughout the entire plane.
Derivative
In the context of complex functions, the derivative is a measure of how a function changes as its input changes, just as in real-valued functions.
However, the derivative in the complex sense must satisfy the Cauchy-Riemann equations.To determine where our function \( w = \cosh \pi z \) is not conformal, we look at its derivative: \( \frac{dw}{dz} = \pi \sinh \pi z \).
The function will not be conformal where this derivative equals zero.
  • To find these points, solve \( \pi \sinh \pi z = 0 \), which reduces to \( \sinh \pi z = 0 \).
  • This occurs when \( \pi z = n\pi i \), or just \( z = ni \) for integers \( n \).
At these points, the function is not conformal because the derivative becomes zero, leading to a failure to preserve angles at that point.

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Most popular questions from this chapter

Find the branch points and the number of sheets of the Riemann surface. $$5+\sqrt[3]{2 x+1}$$

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