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If u(x,y)=e^{x}\sin y\, find f(x,y)=u(x,y)+iv(x,y)\, and check if it satisfies the Cauchy-Riemann equations.

The Cauchy-Riemann equations are u_{x}=v_{y},v_{x}=-u_{y}\,.

u_{x}=e^{x}\sin y\,, u_{y}=e^{x}\cos y\,

v_{y}=e^{x}\sin y\,

v=-e^{x}\cos y+g(x)\,

v_{x}=-e^{x}\cos y+g'(x)\,

For the CR equations to hold, we must have g'(x)=0\, so that g(x)=c\in {\mathbb  {R}}\,.

f(x,y)=e^{x}\sin y+i(-e^{x}\cos y+c)\,

=e^{x}(\sin y-i\cos y)+ic\,

=-ie^{x}(\cos y+{\frac  {1}{-i}}\sin y)+ic\,


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