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...


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

Become a Study.com member to unlock this answer! Create your account

View this answer

Part a.


{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) =...

See full answer below.

Learn more about this topic:

What are Harmonics? - Definition & Types


Chapter 16 / Lesson 50

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.

Related to this Question

Explore our homework questions and answers library