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Unformatted text preview: 13.4 Cauchy-Riemann Equations. Laplaces Equation. The Cauchy-Riemann equations are a system of two linear first-order par- tial differential equations which provide a criterion for differentiability of a complex function: Theorem 2 If a complex function f ( x + iy ) = u ( x, y ) + iv ( x, y ) is holomor- phic in U C , then the first partial derivatives of u and v exist and they satisfy the Cauchy-Riemann equations u x = v y , u y = v x , in U. (609) Theorem 3 If two real-valued continuous functions u, v : R 2 R have continuous first partial derivatives that satisfy the Cauchy-Riemann equations (609) in some domain U C , then the complex function f ( x + iy ) := u ( x, y )+ iv ( x, y ) is holomorphic in U . Remark: valid under weaker conditions (Looman, 1923; Menchoff, 1936) Sketch of Proofs: Theorem 2: For a point z U , we know that the derivative of f is given by f ( z ) = lim z z f ( z ) f ( z ) z z (610) (and that this limit exists). We choose two particular paths z z such that f ( x + iy ) = lim x x u ( x, y ) u ( x , y ) x x + i lim x x v ( x, y ) v ( x , y ) x x , f ( x + iy ) = i lim y y u ( x , y ) u ( x , y ) y y + lim y y v ( x , y ) v ( x , y ) y y . From these equations, we conclude that the first partial derivatives of u, v exist at ( x , y ) and also that u x ( x , y ) = v y ( x , y ) , u y ( x , y ) = v x ( x , y ) . (611) This is true for any point z = x + iy U . Remark: We also conclude from this proof that f ( x + iy ) = u x ( x , y )+ iv x ( x , y ) = iu y ( x , y )+ v y ( x , y ) . (612) 103 Theorem 3: Choose z U . Because U is open, it contains an open ball around z = ( x , y ), and therefore we may find a second point z = ( x, y ) U such that the line segment which connects the two points lies entirely in U . Because of the differentiability of u and v , we may apply the mean value theorem in R 2 . It guarantees the existence of points m 1 , m 2 R 2 on the line segment connecting the points (...
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