MATH 241 Final: MATH 241 UPenn 241sp14Final Exam
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Please clear your desk and put away all notes, books, electronic devices, etc. No outside material is allowed during the exam. Make sure to clearly indicate your responses; box your nal answers if necessary. My signature below certi es that i will comply with the university of penn- sylvania"s code of academic integrity in completing this examination. Facts about bessel functions: bessel"s equation: r2f (r) + rf (r) + ( 2r2 m2)f (r) = 0 for each integer m 0. The only solutions which are bounded at r = 0 are f (r) = cjm( r) for a constant c: orthogonality relation: writing zmn as the nth zero of jm(z), 0 rjn(zmnr)jn(zmkr) dr = 0, n 6= k for each m = 0, 1, . Table of fourier transform pairs ( , > 0) u(x) = f 1[u ] e x2 e |x| u(x) =(0. 2 u(x) = f 1[u ] r e x.
