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Chapter 13: Problem 94

In each part, show that \(u(x, y)\) and \(v(x, y)\) satisfy the Cauchy-Riemann equations $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \quad \text { and } \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$ $$\begin{array}{ll}{\text { (a) } u=x^{2}-y^{2},} & {v=2 x y} \\ {\text { (b) } u=e^{x} \cos y,} & {v=e^{x} \sin y} \\ {\text { (c) } u=\ln \left(x^{2}+y^{2}\right),} & {v=2 \tan ^{-1}(y / x)}\end{array}$$

Short Answer

Expert verified
All three pairs (u,v) satisfy the Cauchy-Riemann equations.

Step by step solution

01

Calculate Partial Derivatives for (a)

Given \( u = x^2 - y^2 \) and \( v = 2xy \).Calculate \( \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(x^2 - y^2) = 2x \).Calculate \( \frac{\partial v}{\partial y} = \frac{\partial}{\partial y}(2xy) = 2x \).Calculate \( \frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(x^2 - y^2) = -2y \).Calculate \( \frac{\partial v}{\partial x} = \frac{\partial}{\partial x}(2xy) = 2y \).Verify both Cauchy-Riemann equations:1. \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) : \( 2x = 2x \) (True)2. \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \) : \( -2y = -2y \) (True).
02

Calculate Partial Derivatives for (b)

Given \( u = e^x \cos y \) and \( v = e^x \sin y \).Calculate \( \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(e^x \cos y) = e^x \cos y \).Calculate \( \frac{\partial v}{\partial y} = \frac{\partial}{\partial y}(e^x \sin y) = e^x \cos y \).Calculate \( \frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(e^x \cos y) = -e^x \sin y \).Calculate \( \frac{\partial v}{\partial x} = \frac{\partial}{\partial x}(e^x \sin y) = e^x \sin y \).Verify both Cauchy-Riemann equations:1. \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) : \( e^x \cos y = e^x \cos y \) (True)2. \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \) : \( -e^x \sin y = -e^x \sin y \) (True).
03

Calculate Partial Derivatives for (c)

Given \( u = \ln(x^2 + y^2) \) and \( v = 2 \tan^{-1}(\frac{y}{x}) \).Calculate \( \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(\ln(x^2 + y^2)) = \frac{2x}{x^2 + y^2} \).Calculate \( \frac{\partial v}{\partial y} = \frac{\partial}{\partial y}(2 \tan^{-1}(\frac{y}{x})) = \frac{2x}{x^2 + y^2} \) using chain rule.Calculate \( \frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(\ln(x^2 + y^2)) = \frac{2y}{x^2 + y^2} \).Calculate \( \frac{\partial v}{\partial x} = \frac{\partial}{\partial x}(2 \tan^{-1}(\frac{y}{x})) = -\frac{2y}{x^2 + y^2} \) using chain rule.Verify both Cauchy-Riemann equations:1. \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) : \( \frac{2x}{x^2 + y^2} = \frac{2x}{x^2 + y^2} \) (True)2. \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \) : \( \frac{2y}{x^2 + y^2} = \frac{2y}{x^2 + y^2} \) (True).

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Key Concepts

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

Complex Analysis
Complex analysis is a fascinating branch of mathematics focused on complex numbers and the functions that use them. A core idea in complex analysis is the concept of holomorphic functions, which are complex functions that are differentiable at every point in their domain. This differentiability is similar to the concept of differentiation in calculus but is more stringent in terms of requirements. For a function to be holomorphic, it must satisfy the Cauchy-Riemann equations, which serve as a bridge connecting the real and imaginary parts of a complex function.
  • The real part of the function is denoted as \( u(x,y) \), and the imaginary part is \( v(x, y) \).
  • The function expressed as \( f(z) = u(x, y) + iv(x, y) \) must satisfy \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).
These equations ensure that the function behaves nicely and aligns with the properties of complex numbers, emphasizing symmetry and continuity. Verification of these equations often involves partial derivatives, illustrating the interplay between calculus and complex variables.
Partial Derivatives
In the realm of calculus, partial derivatives hold a significant role, especially when dealing with functions of more than one variable. Partial derivatives capture the idea of differentiation with respect to one variable while keeping others constant. This becomes crucial when dealing with complex functions, which are inherently expressed through multiple variables:
  • For a function \( u(x, y) \), the partial derivative with respect to \( x \) is denoted as \( \frac{\partial u}{\partial x} \).
  • Similarly, the partial derivative with respect to \( y \) is written as \( \frac{\partial u}{\partial y} \).
These derivatives are essential to establishing the Cauchy-Riemann conditions, which require computing these derivatives to verify if a function is holomorphic. The process involves systematically taking the derivative and comparing these to see if the Cauchy-Riemann equations hold true. Approaching this methodically ensures clarity and accuracy in verifying complex functions.
Verification of Equations
Verification of equations, especially in complex analysis, is essential for validating whether a function is holomorphic. This process involves more than just computation; it requires a deeper understanding of the relationships between the function's real and imaginary components:
  • First, compute the partial derivatives \( \frac{\partial u}{\partial x} \), \( \frac{\partial v}{\partial y} \), \( \frac{\partial u}{\partial y} \), and \( \frac{\partial v}{\partial x} \).
  • Next, check if these satisfy the Cauchy-Riemann equations, confirming if \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).
Successfully verifying these conditions means the complex function behaves in a predictable, smooth manner over its domain. Each part of the solution exemplifies this process, reinforcing the verification of whether each function meets these critical equations. Such practices emphasize the importance of analytical skills in complex analysis and the beauty of mathematical symmetry.

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Most popular questions from this chapter

Given that the functions \(u=u(x, y, z), v=v(x, y, z)\) \(w=w(x, y, z),\) and \(f(u, v, w)\) are all differentiable, show that $$ \nabla f(u, v, w)=\frac{\partial f}{\partial u} \nabla u+\frac{\partial f}{\partial v} \nabla v+\frac{\partial f}{\partial w} \nabla w $$

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