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Unformatted text preview: 13.4 CauchyRiemann Equations. Laplace’s Equation. The CauchyRiemann equations are a system of two linear firstorder 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 CauchyRiemann equations u x = v y , u y = − v x , in U. (609) Theorem 3 If two realvalued continuous functions u, v : R 2 → R have continuous first partial derivatives that satisfy the CauchyRiemann 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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This note was uploaded on 11/17/2011 for the course MATH 529 taught by Professor Staff during the Spring '08 term at UNC.
 Spring '08
 Staff
 Math, Equations

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