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nick1337 · Accepted Answer

We are given the function f(z)=eiz and we need to verify that its real and imaginary parts satisfy the Cauchy-Riemann equations, and then deduce the analyticity of f(z).Let z=x+iy be a complex variable. Then we can write:f(z)=eiz=ei(x+iy)=e−y(cosx+isinx)So the real part of f(z) is:u(x,y)=e−ycosxand the imaginary part of f(z) is:v(x,y)=e−ysinxNow, we need to check if these functions satisfy the Cauchy-Riemann equations. According to the Cauchy-Riemann equations, if a complex function f(z) is differentiable at a point z0=x0+iy0, then its real and imaginary parts satisfy the following partial differential equations:∂u∂x=∂v∂y∂u∂y=−∂v∂xLet&#039;s verify these equations for u(x,y) and v(x,y):∂u∂x=−e−ysinx=∂v∂y∂u∂y=−e−ycosx=−∂v∂xWe can see that both equations are satisfied. Therefore, the real and imaginary parts of f(z) satisfy the Cauchy-Riemann equations.Since the real and imaginary parts of f(z) are both continuously differentiable and satisfy the Cauchy-Riemann equations, we can conclude that f(z) is analytic in the entire complex plane.

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