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Chapter 7: Problem 14

Consider the function \(u(x, y)=x^{3}-3 x y^{2}\). a. Show that \(u(x, y)\) is harmonic; that is, \(\nabla^{2} u=0\) b. Find its harmonic conjugate, \(v(x, y)\). c. Find a differentiable function, \(f(z)\), for which \(u(x, y)\) is the real part. d. Determine \(f^{\prime}(z)\) for the function in part c. [Use \(f^{\prime}(z)=\frac{\partial_{2}}{\partial x}+i \frac{\partial v}{\partial x}\) and rewrite your answer as a function of \(z .]\)

Short Answer

Expert verified
a. The function \(u(x, y) = x^3 - 3xy^2\) is a harmonic function because \(∇^2 u = 0\). b. Its harmonic conjugate is \(v(x, y) = y^3 - 3x^2y + c\). c. The differentiable function is then \(f(z) = z^3 + c\). d. The derivative of this function is \(f^{\prime}(z) = 3z^2\).

Step by step solution

01

Prove \(u(x, y)\) is harmonic

To check if \(u(x, y)\) is harmonic, we calculate the Laplacian, \(∇^2 u\). This is the sum of the second partial derivatives of u with respect to x and y: \[∇^2 u = \frac{∂^2u}{∂x^2} + \frac{∂^2u}{∂y^2}\]
02

Calculate the Laplacian

Upon calculating, we get: \[∇^2 u = \frac{∂^2u}{∂x^2} + \frac{∂^2u}{∂y^2} = 6x-6x = 0\] as both terms cancel out, proving that \(u(x, y)\) is indeed harmonic.
03

Find the harmonic conjugate

The harmonic conjugate \(v(x, y)\) needs to satisfy the Cauchy-Riemann equations: \[\frac{∂u}{∂x} = \frac{∂v}{∂y} \quad and \quad -\frac{∂u}{∂y} = \frac{∂v}{∂x}\] Now let's solve these equations.
04

Calculate the harmonic conjugate

After solving the Cauchy-Riemann equations, the harmonic conjugate is found to be \(v(x, y) = y^3 - 3x^2y + c\), where c is the constant of integration.
05

Find the differentiable function, \(f(z)\)

To find the complex function \(f(z)\), we combine the original function \(u(x, y)\) and its conjugate \(v(x, y)\) to get: \(f(z) = u(x, y) + iv(x, y) = x^3 - 3xy^2 + i(y^3 - 3x^2y + c)\). This is the intended complex function. Rewriting z as \(x + iy\), we get: \(f(z) = z^3 + c\).
06

Determine \(f^{\prime}(z)\)

The derivative of \(f(z)\) is found by using the definition: \(f^{\prime}(z)=\frac{∂}{∂x} + i \frac{∂v}{∂x}\). Upon doing the calculus, we get: \(f^{\prime}(z) = 3z^2\).

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Most popular questions from this chapter

. For the following, determine if the given point is a removable singularity, an essential singularity, or a pole (indicate its order). a. \(\frac{1-\cos z}{z^{2}}, \quad z=0\) b. \(\frac{\sin 2}{z^{2}}, \quad z=0\) c. \(\frac{z^{2}-1}{(z-1)^{2}}, \quad z=1\). d. \(z e^{1 / z}, \quad z=0\). e. \(\cos \frac{\pi}{\pi-\pi}, \quad z=\pi\)

Write the following in rectangular form, \(z=a+i b\). a. \(4 e^{i \pi / 6}\) b. \(\sqrt{2} e^{5 i \pi / 4}\) c. \((1-i)^{100} .\)

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