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extra_5 - Extra Problems Sheet 5 Stephen Taylor June 6 2005...

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Extra Problems Sheet 5 Stephen Taylor June 6, 2005 1. The notion of analyticity requires that the function f ( x, y ) = u ( x, y )+ iv ( x, y ) can be written in terms of z = x + iy alone, without using ¯ z = x - iy . To make this more explicit, we can introduce the change of variables ξ = x + iy η = x - iy or x = ( ξ + η ) / 2 y = ( ξ - η ) / 2 i producing the function ˜ f ( ξ, η ) f ( x ( ξ, η ) , y ( ξ, η )) (a) Using the chain rule, show formally that ˜ f ∂ξ = 1 2 ∂u ∂x + ∂v ∂y + i 2 ∂v ∂x - ∂u ∂y ˜ f ∂η = 1 2 ∂u ∂x - ∂v ∂y + i 2 ∂v ∂x + ∂u ∂y We first note that ˜ f ∂ξ = ∂ξ u ξ + η 2 , ξ - η 2 i + iv ξ + η 2 , ξ - η 2 i applying the chain rule we find ˜ f ∂ξ = 1 2 u x + 1 2 i u y + i 1 2 v x + 1 2 i v y = 1 2 ( u x + v y ) + i 2 ( v x - u y ) Similarly we find ˜ f ∂η = ∂η u ξ + η 2 , ξ - η 2 i + iv ξ + η 2 , ξ - η 2 i again applying the chain rule we find ˜ f ∂η = 1 2 u x - 1 2 i u y + i 1 2 v x - 1 2 i v y 1
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= 1 2 ( u x - v y ) + i 2 ( v x + u y ) and we have shown the desired result.
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