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+ ivof a complex variable z= x+ iy, defined in some region of the complex plane, where u, v, x, yare real. That is to say, ( )( ,)( ,),fzu x yiv 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( )()( )limzdfzf zzfzdzzΔ →+ Δ−=Δis 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 notjust mean that the function is reasonably smooth. The crucial difference from a function of a real variable is that Δzcan approach zero from any directionin 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 xiyf xiyxiy∂+∂+=∂∂which we can write in terms of u,v: ( ,)( ,)( ,)( ,.()()u x yv x yu x yv x yii)xxiyiy∂∂∂∂+=+∂∂∂∂Equating real and imaginary parts of this equation we find: , .uvvuxyxy∂∂∂∂== −∂∂∂∂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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