CV10

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If u(x,y) = e^x\sin y\, find f(x,y) = u(x,y) + i v(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\isin\mathbb{R}\,.

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

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

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

=-i e^x e^{iy} + ic = -i e^z+ic\,

Main Page : Complex Variables

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