Problem 4: Conditional Expressions Mathematica is pretty much almighty when dealing with integrals. However, there are times when you are expecting a simple expression, Mathematica instead gives you a somewhat complicated ConditionalExpression. Sometimes those conditions are pedantic and obvious. But sometimes those can be an indicator for subtleties. (a) Ask Mathematica to calculate the following integral -ix dk **2 + m2 (3) with (i) no conditions on m and and (ii) assuming m >0 and 1 € R. (b) Ask Mathematica to calculate the following Fourier Transform by directly integrating eik sin(I) Lda dr (4) T with (i) no conditions on k and (ii) |k|1. (c) Verify the integrand is analytic for I E R (Is there any particular point should we pay extra attention?). (d) To understand the behavior of the above integral in part (b), one has to resort to complex analysis. Here is one possible approach. Motivated by the observation in part (c), one consider a slightly modified problem eiks sin(I) L du (5) 2-ie with the understanding that € +0+ at the end of the calculation (i) Explain the behavior of the integral as a function of k. i.e. the answers in part (b) (ii) and (iii) by constructing appropriate contour(s) when evaluating equation (5). (d) Evaluate the inverse Fourier Transform by directly integrating 1 dke tkr f(k) (6) 211 where f(K) if kl 1 Is this answer consistent with the result in part (b) and (c)?
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