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# ps03 - Physics 651 Problem Set 3 Read Chapter 1(due 1 Show...

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Physics 651 – Problem Set 3 (due 9/22/03) Read Chapter 1. 1. Show that in the sense of distribution theory lim ± +0 1 x + = P 1 x - iπ δ ( x ) , where P is the principal value. To prove this identity, consider integrals of both sides of the equation with an arbitrary, smooth test function. 2. Deﬁne the Fourier transform of a function f ( x ) to be e f ( k ) = Z dx f ( x ) e - ikx . a) Evaluate the Fourier transform of θ ( x ). (Hint: Regulate the integral at inﬁnity by replacing e - ikx with e - ( ik + ± ) x , taking the limit ± +0 at the end.) c) Compute the Fourier transform of 1 / ( x 2 + a 2 ) for real a by closing the integration contour in the upper or lower complex x -plane. 3. Let Z[J] be the generating functional of all correlation functions, or Feynman dia- grams, of the λφ 4 theory. However, some diagrams compose of disconnected parts. It turns out that ln Z[J] is the generating functional of connected diagrams or cor- relation functions. To check this fact, work out up to the leading

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ps03 - Physics 651 Problem Set 3 Read Chapter 1(due 1 Show...

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