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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 0 "" }{TEXT -1 54 "PHYS 312
:\nSOLUTIONS TO PROBLEMS\nSect 5.6\nProblem 5.6:1" }}{PARA 256 "" 0 "
" {TEXT 257 518 "The infinite series representation for the Bessel fun
ction is\n\\[J_n(x)=(\\frac\{x\}\{2\})^n\\sum_\{m=0\}^\{\\infty\}\\fra
c\{(-1)^m(\\frac\{x\}\{2\})^\{2m\}\}\{m!(n+m)!\}\\]\nHence\n\\[\\frac
\{d\}\{dx\}(x^\{-n\}J_n(x))=(\\frac\{1\}\{2\})^n\\sum_\{m=1\}^\\infty
\n\\frac\{(-1)^m(\\frac\{x\}\{2\})^\{2m-1\}\}\{(m-1)!(n+m)!\}\\]\nWe p
ut $k=m-1$\n\\[\\frac\{d\}\{dx\}(x^\{-n\}J_n(x))=-x^\{-n\}(\\frac\{x\}
\{2\})^n\\sum_\{k=0\}^\\infty\n\\frac\{\\frac\{x\}\{2\}(\\frac\{x\}\{2
\})^\{2k\}(-1)^k\}\{k!(n+1+k)!\}\\]\n\\[=-x^\{-n\}(\\frac\{x\}\{2\})^
\{n+1\}\\sum_\{k=0\}^\\infty\\frac\{(-1)^k(\\frac\{x\}\{2\})^\{2k\}\}
\{k!(n+1+k)!\}=-x^\{-n\}J_\{n+1\}(x)\\]\n" }{TEXT -1 1 "\n" }}}}{MARK 
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