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Let u = u(x,y) = e^x (x\cos x - y\sin y) . a. Show that u = u(x,y) is a harmonic function....

Question:

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

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

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

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

Find The Harmonic Conjugate :

To find the given function is harmonic function if its satisfy the following condition

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

To find the harmonic conjugate we evaluate the following integral (By Milne's Method)

{eq}f(z)=\int \left [\phi _1(z,0) -i\phi _2(z,0) \right ]dz+c\\ {/eq}

Answer and Explanation:

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What are Harmonics? - Definition & Types

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Chapter 16 / Lesson 50
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