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Unformatted text preview: Lecture 08 Define u : C R C by u ( z,t ) = exp 2 zt 1 2 z 2 1 2 t 2 . Theorem 1. U : V S V F is given by ( Uf )( z ) = 1 Z R u ( z,t ) f ( t ) dt for f V S and z C . Proof. It is enough to check f = H n ; that is, check 1 Z R exp 2 zt 1 2 z 2 1 2 t 2 H n ( t ) ( t ) dt = 2 n 2 z n . Multiply by s n n ! and sum. 1 Z R X n =0 H n ( t ) s n n ! exp 2 zt 1 2 z 2 t 2 dt = 1 Z R exp 2 ts s 2 + 2 zt 1 2 z 2 t 2 dt. (1) Complete the square in the exponent: t s + z 2 2 = t 2 + s 2 + 1 2 z 2 + 2 sz 2 st + 2 tz, 1 therefore t 2 s 2 1 2 z 2 + 2 tz + 2 st = 2 sz t s + z 2 2 , and continuing from equation (1) = 1 Z R exp { 2 sz } exp t s + z 2 2 dt = X n =0 2 n z n s n n ! . Equate coefficients of s n n ! . Remark 1. We used the fact that if a C then Z R exp { ( t a ) 2 } dt = exp { a 2 } Z R exp { 2 at t 2 } dt = ....
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 Summer '10
 Staff
 Algebra

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