conformal

# conformal - (because f z is analytic and conformal Applying...

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ACM95a/100a Lecture Notes Niles A. Pierce Caltech 2008 Open mapping property If f ( z ) is analytic in a domain D , the image of D under the mapping w = f ( z ) is a domain in the w plane (i.e. open connected sets map to open connected sets). If the image of the boundary is a simple closed curve, the image of the domain can be located simply by checking one point not on the boundary. Inverse mappings If f ( z ) is analytic at z 0 with f 0 ( z 0 ) 6 = 0 then w = f ( z ) has a unique analytic inverse z = F ( w ) in the neighborhood of w 0 with F 0 ( w ) = 1 /f 0 ( z ). Hence, F 0 ( w ) 6 = 0 and the inverse mapping z = F ( w ) is locally conformal. Mapping harmonic functions Suppose that the analytic function w = f ( z ) = u ( x,y ) + iv ( x,y ) conformally maps D z to D w and consider a function φ z ( x,y ) that is harmonic in D z with continuous ﬁrst and second partials (e.g. it is the real or imaginary part of a function that is analytic in D z ). In D w we can then deﬁne the function φ w ( u,v ) φ z ( x ( u,v ) ,y ( u,v )) (1) using the fact that the inverse function z = F ( w ) = x ( u,v ) + iy ( u,v ) is a unique analytic function
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Unformatted text preview: (because f ( z ) is analytic and conformal). Applying the chain rule and using the Cauchy-Riemann equations, this function can be shown to satisfy ∂ 2 φ z ( x,y ) ∂x 2 + ∂ 2 φ z ( x,y ) ∂y 2 = ∂ 2 φ w ( u,v ) ∂u 2 + ∂ 2 φ w ( u,v ) ∂v 2 ! | f ( z ) | 2 = 0 . (2) Since f is conformal in D z , f ( z ) 6 = 0, so φ w ( u,v ) is harmonic in D w . The Dirichlet problem Boundary conditions are required to obtain a unique solution to the Laplace equation. The Dirich-let problem is obtained by solving the Laplace equation with φ speciﬁed on the boundary. Potential Flow Over a Joukowski Airfoil-2-1 1 2-2-1. 5-1-0.5 0.5 1 1.5 2 u v Stream function in the w plane-2-1 1 2-2-1.5-1-0.5 0.5 1 1.5 2 x y Stream function in the z plane-2-1 1 2-2-1.5-1-0.5 0.5 1 1.5 2 a b Stream function in the c plane...
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