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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 √ π Z R u ( z,t ) f ( t ) dt for f ∈ V S and z ∈ C . Proof. It is enough to check f = H n φ ; that is, check 1 √ π Z R exp √ 2 zt 1 2 z 2 1 2 t 2 H n ( t ) φ ( t ) dt = 2 n 2 z n . Multiply by s n n ! and sum. 1 √ π Z R ∞ X n =0 H n ( t ) s n n ! exp √ 2 zt 1 2 z 2 t 2 dt = 1 √ π 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 √ π 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 = √ π....
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This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.
 Summer '10
 Staff
 Algebra

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