Let u=u(x,y)=e^x(xcos x-ysin y). a. Show that u=u(x,y) is a harmonic function. b. Find the...

Question:

Let {eq}\displaystyle u=u(x,y)=e^x(x\cos x-y\sin y). {/eq}

a. Show that {eq}\displaystyle u=u(x,y) {/eq} is a harmonic function.

b. Find the harmonic conjugate of {eq}\displaystyle u=u(x,y). {/eq}

c. Write the analytic function {eq}\displaystyle f(z)=u(x,y)+v(x,y)i {/eq} in terms of {eq}\displaystyle z. {/eq}

Harmonic function:

A function {eq}u(x,y) {/eq} is called harmonic if it satisfies the following partial differential equation:

{eq}\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0 {/eq}

Answer and Explanation:

{eq}\textbf{Consider}\\ u(x,y)=e^x(x\cos x-y\sin y)\\ \frac{\partial u}{\partial x}=e^x\left(x\cos \left(x\right)-y\sin...

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What are Harmonics? - Definition & Types

from Intro to Humanities: Tutoring Solution

Chapter 16 / Lesson 50
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