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Question Number 204226 by MathedUp last updated on 09/Feb/24
God Damn it why my Integration Test is Wrong?  Σ_(h=0) ^∞  h∙J_ν (h) is Conv??  Laplace transform ∫_0 ^∞  e^(−st) ∙  We already Know ∫_0 ^∞  e^(−st) J_ν (t)dt=(1/( (√(s^2 +1))(s+(√(s^2 +1)))^ν ))  ((d  )/ds) ∫_0 ^∞  e^(−st) J_ν (t)dt=((d  )/ds) (1/( (√(s^2 +1))(s+(√(s^2 +1)))^𝛎 ))  −∫_0 ^∞  te^(−st) J_ν (t)dt=−((s+ν(√(s^2 +1)))/( (√((s^2 +1)^3 ))(s+(√(s^2 +1)))^ν ))   (Thx Wolfram)  ∴∫_0 ^∞  te^(−st) J_ν (t)dt=((s+ν(√(s^2 +1)))/((s^2 +1)(√(s^2 +1))(s+(√(s^2 +1)))^ν ))  s→0   ∫_0 ^∞  tJ_ν (t)dt=ν     Σ_(h=0) ^∞  h∙J_ν (h) Div....    :(
$$\mathrm{God}\:\mathrm{Damn}\:\mathrm{it}\:\mathrm{why}\:\mathrm{my}\:\mathrm{Integration}\:\mathrm{Test}\:\mathrm{is}\:\mathrm{Wrong}? \\ $$$$\underset{{h}=\mathrm{0}} {\overset{\infty} {\sum}}\:{h}\centerdot{J}_{\nu} \left({h}\right)\:\mathrm{is}\:\mathrm{Conv}?? \\ $$$$\mathrm{Laplace}\:\mathrm{transform}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{st}} \centerdot \\ $$$$\mathrm{We}\:\mathrm{already}\:\mathrm{Know}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{st}} {J}_{\nu} \left({t}\right)\mathrm{d}{t}=\frac{\mathrm{1}}{\:\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\left({s}+\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\right)^{\nu} } \\ $$$$\frac{\mathrm{d}\:\:}{\mathrm{d}{s}}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{st}} {J}_{\nu} \left({t}\right)\mathrm{d}{t}=\frac{\mathrm{d}\:\:}{\mathrm{d}{s}}\:\frac{\mathrm{1}}{\:\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\left({s}+\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\right)^{\boldsymbol{\nu}} } \\ $$$$−\int_{\mathrm{0}} ^{\infty} \:{te}^{−{st}} {J}_{\nu} \left({t}\right)\mathrm{d}{t}=−\frac{{s}+\nu\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}}{\:\sqrt{\left({s}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} }\left({s}+\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\right)^{\nu} }\: \\ $$$$\left(\mathrm{Thx}\:\mathrm{Wolfram}\right) \\ $$$$\therefore\int_{\mathrm{0}} ^{\infty} \:{te}^{−{st}} {J}_{\nu} \left({t}\right)\mathrm{d}{t}=\frac{{s}+\nu\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}}{\left({s}^{\mathrm{2}} +\mathrm{1}\right)\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\left({s}+\sqrt{{s}^{\mathrm{2}} +\mathrm{1}}\right)^{\nu} } \\ $$$${s}\rightarrow\mathrm{0} \\ $$$$\:\int_{\mathrm{0}} ^{\infty} \:{tJ}_{\nu} \left({t}\right)\mathrm{d}{t}=\nu \\ $$$$\:\:\:\underset{{h}=\mathrm{0}} {\overset{\infty} {\sum}}\:{h}\centerdot{J}_{\nu} \left({h}\right)\:\mathrm{Div}….\:\:\:\::\left(\right. \\ $$

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