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ComplexVariable - Functions of a Complex Variable Contour...

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Functions of a Complex Variable: Contour Integration and Steepest Descent Michael Fowler 9/19/08 Analytic Functions Suppose we have a complex function f = u + iv of a complex variable z = x + iy , defined in some region of the complex plane, where u , v , x , y are real. That is to say, ( ) ( , ) ( , ), f z u x y iv x y = + with u ( x , y ) and v ( x , y ) real functions in the plane. We now assume that in this region f ( z ) is differentiable, that is to say, 0 ( ) ( ) ( ) lim z df z f z z f z dz z Δ → + Δ = Δ is well-defined. What does this tell us about the functions u ( x , y ) and v ( x , y ), the real and imaginary parts of f ( z )? In fact, the property of differentiability for a function of a complex variable tells us a lot! It does not just mean that the function is reasonably smooth. The crucial difference from a function of a real variable is that Δ z can approach zero from any direction in the complex plane, and the limit in these different directions must be the same . Of course, there are only two independent directions, so what we are really saying is ( ) ( ) , ( ) f x iy f x iy x iy + + = which we can write in terms of u , v : ( , ) ( , ) ( , ) ( , . ( ) ( ) u x y v x y u x y v x y i i ) x x iy iy + = + Equating real and imaginary parts of this equation we find: , . u v v u x y x y = = − These are called the Cauchy-Riemann equations . It immediately follows that both u ( x , y ) and v ( x , y ) must satisfy the two-dimensional Laplacian
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