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Q-use-the-parseval-relation-of-hankel-transfrom-to-evaluate-the-Integral-0-J-1-ar-J-1-br-r-for-gt-1-2-0-lt-a-lt-b-where-J-n-x




Question Number 186531 by normans last updated on 05/Feb/23
    Q.use the parseval relation of hankel transfrom to evaluate the Integral                                  ∫_0 ^∞   ((J_(𝛄+1) (ar)J_(𝛄+1) (br))/r) ,  for 𝛄>−(1/2) ,  0<a<b     where J_n (x) are bessel funtions.
$$ \\ $$$$\:\:\mathbb{Q}.\boldsymbol{{use}}\:\boldsymbol{{the}}\:\boldsymbol{{parseval}}\:\boldsymbol{{relation}}\:\boldsymbol{{of}}\:\boldsymbol{{hankel}}\:\boldsymbol{{transfrom}}\:\boldsymbol{{to}}\:\boldsymbol{{evaluate}}\:\boldsymbol{{the}}\:\boldsymbol{{Integral}}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{\infty} {\int}_{\mathrm{0}} \:\:\frac{\boldsymbol{{J}}_{\boldsymbol{\gamma}+\mathrm{1}} \left(\boldsymbol{{ar}}\right)\boldsymbol{{J}}_{\boldsymbol{\gamma}+\mathrm{1}} \left(\boldsymbol{{br}}\right)}{\boldsymbol{{r}}}\:,\:\:\boldsymbol{{for}}\:\boldsymbol{\gamma}>−\frac{\mathrm{1}}{\mathrm{2}}\:,\:\:\mathrm{0}<\boldsymbol{{a}}<\boldsymbol{{b}} \\ $$$$\:\:\:\boldsymbol{{where}}\:\boldsymbol{{J}}_{\boldsymbol{{n}}} \left(\boldsymbol{{x}}\right)\:\boldsymbol{{are}}\:\boldsymbol{{bessel}}\:\boldsymbol{{funtions}}. \\ $$$$ \\ $$

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