# hwk5 - r real and positive by analytic continuation....

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Physics 604 Homework #5 Fall ‘11 Dr. Drake 1. Arfken Chapter 7 : 3.4, 3.5 2. Evaluate the following integral Z 1 0 dte ixt 2 , for large x with x real and positive. You will ﬁnd that the lowest order term scales as x - 1 / 2 and the next order as e ix /x . Calculate both terms. Hint: The real axis is not the best contour of integration. 3. Evaluate the following integral for x large and real Z π 0 dte ix cos t . 4. Consider the following integral I ( z ) = Z 0 dte - zt ln ( t ) with - π < Arg ( t ) < π . (a) For what values of complex z is the integral deﬁned? (b) Evaluate I ( re - ) with
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Unformatted text preview: r real and positive by analytic continuation. Ex-press your answer in terms of I ( r ). 5. (challenge problem: not for grading) Find the analytic continuation of K ν ( z ): K ν ( re iπ ) = e-iνπ K ν ( r )-iπI ν ( r ) with r real and positive and K ν ( z ) = 1 2 Z ∞ e-z 2 ( s + 1 s ) s 1+ ν . Hint: Let z = re iθ and let θ increase from 0 to π . As you do this make sure that the integral remains bounded both around s = 0 and ∞ ....
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## This note was uploaded on 12/30/2011 for the course PHYSICS 604 taught by Professor Drake during the Fall '11 term at Maryland.

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