MODULAR FORMS-page71

MODULAR FORMS-page71 - 67 Use the function q = e2i to map...

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67 Use the function q = e 2 πiτ to map the segment { τ : | Re τ | ≤ 1 2 , Im τ = h } onto the circle C : | q | = e - 2 πh . When we move along the segment from the point 1 2 + ih to the point - 1 2 + ih the image point moves along the circle in the clockwise way. We have 1 2 πi Z ∂P 1 f 0 f = - 1 2 πi Z C (2 πiν ( f ) q ν ( f ) + ... ) dq 2 πiq ( a ν ( f ) q ν ( f ) + ... ) = - ν ( f ) . If we integrate along the part ∂P 2 of the boundary of P which lies on the circle C r ( ρ 2 ) we get lim r 0 1 2 πi Z ∂P 2 f 0 f = - 1 6 ν ρ 2 ( f ) . This is because the arc ∂P 2 approaches to the one-sixth of the full circle when its radius goes to zero. Also we take into account that the direction of the path is clockwise. Similarly, if we let ∂P 3 = ∂P C r ( i ) ,∂P 4 = ∂P C r ( ρ ), we find lim r 0 1 2 πi Z ∂P 3 f 0 f = - 1 2 ν i ( f ) . lim
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This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.

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