(a) Prove that u = e^-x (x sin y - y cos y) is harmonic. (b) Find v such that f(z) = u + iv is analytic.

Question

(a) Prove that [latex]u = e^-x (x \sin y - y \cos y)[/latex] is harmonic.

(b) Find v such that [latex] f(z) = u + iv [/latex] is analytic.

Accepted Answer

(a)

[latex] \begin{aligned} \frac{\partial u}{\partial x} &=\left(e^{-x} ight)(\sin y)+\left(-e^{-x} ight)(x \sin y-y \cos y)=e^{-x} \sin y-x e^{-x} \sin y+y e^{-x} \cos y \ \frac{\partial^2 u}{\partial x^2} &=\frac{\partial}{\partial x}\left(e^{-x} \sin y-x e^{-x} \sin y+y e^{-x} \cos y ight)=-2 e^{-x} \sin y+x e^{-x} \sin y-y e^{-x} \cos y &(1)\ \frac{\partial u}{\partial y} &=e^{-x}(x \cos y+y \sin y-\cos y)=x e^{-x} \cos y+y e^{-x} \sin y-e^{-x} \cos y \ \frac{\partial^2 u}{\partial y^2} &=\frac{\partial}{\partial y}\left(x e^{-x} \cos y+y e^{-x} \sin y-e^{-x} \cos y ight)=-x e^{-x} \sin y+2 e^{-x} \sin y+y e^{-x} \cos y&(2) \end{aligned} [/latex]

Adding (1) and (2) yields [latex] \left(\partial^2 u / \partial x^2 ight)+\left(\partial^2 u / \partial y^2 ight)=0 [/latex] and u is harmonic.

(b) From the Cauchy –Riemann equations,

[latex] \begin{aligned} &\frac{\partial v}{\partial y}=\frac{\partial u}{\partial x}=e^{-x} \sin y-x e^{-x} \sin y+y e^{-x} \cos y &(3)\ &\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}=e^{-x} \cos y-x e^{-x} \cos y-y e^{-z} \sin y &(4) \end{aligned} [/latex]

Integrate (3) with respect to y, keeping x constant. Then

[latex] \begin{aligned} v &=-e^{-x} \cos y+x e^{-x} \cos y+e^{-x}(y \sin y+\cos y)+F(x) \ &=y e^{-x} \sin y+x e^{-x} \cos y+F(x)&(5) \end{aligned} [/latex]

where F(x) is an arbitrary real function of x.
Substitute (5) into (4) and obtain

[latex] -y e^{-x} \sin y-x e^{-x} \cos y+e^{-x} \cos y+F^{\prime}(x)=-y e^{-x} \sin y-x e^{-x} \cos y-y e^{-x} \sin y [/latex]

or [latex] F^{\prime}(x)=0 [/latex] and F(x) = c, a constant. Then, from (5),

[latex] v=e^{-x}(y \sin y+x \cos y)+c [/latex]

For another method, see Problem 3.40.

Question 3.7: (a) Prove that u = e^-x (x sin y - y cos y) is harmonic. (b)...

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