Answer to Question #142498 in Complex Analysis for Doll

Question #142498
Prove that u(x,y) given by the following is harmonic obtain it's corresponding conjugate and original function f(z)
u(x,y)=e^xCosy
1
Expert's answer
2020-11-05T16:09:00-0500

"u_x= e^x cos y=v_y." Hence "v=\\int e^x cos\\ y \\ dy = e^x sin \\ y + \\phi(x)". Hence "v_x= e^x sin y+\\phi^{'}(x)=-u_y= e^x sin \\ y." "\\Rightarrow \\phi^{'}(x)=0\\Rightarrow \\phi(x)=c" , constant.

Hence "v(x,y)= e^x sin \\ y+c." Hence "f(z)=u+iv=e^x(cos\\ y+i \\ sin \\ y)+ic=e^{x+iy}+ic=e^z+ic" .


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