Let u=u(x,y)=e^x(x\cos y-y\sin 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 y-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 functions and Harmonic conjugates:

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

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

A function {eq}v(x,y) {/eq} is a harmonic conjugate of a harmonic function {eq}u(x,y) {/eq} if and only if the following equations are satisfied:

{eq}\dfrac{\partial u}{\partial x} = \dfrac{\partial v}{\partial y} \\ \dfrac{\partial u}{\partial y} = - \dfrac{\partial v}{\partial x} {/eq}

Answer and Explanation: 1

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Part a.

Given

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

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What are Harmonics? - Definition & Types

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Chapter 16 / Lesson 50
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Harmonics are a widely used musical technique, but where do they come from? This lesson will focus on what a harmonic is, where they come from, and how this all relates back to musical pitch.


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