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lim-t-0-C-1-J-t-C-2-Y-t-H-t-C-1-J-t-C-2-Y-t-R-J-z-Bessel-function-First-kind-Y-z-Bessel-function-Second-Kind-H-z-Struve-H-function-

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Question Number 220269 by SdC355 last updated on 10/May/25
lim_(t→0) ((C_1 J_ν (t)+C_2 Y_ν (t)+H_ν (t))/(C_1 J_ν (t)+C_2 Y_ν (t)))=?? ν∈R J_ν (z) Bessel function First kind Y_ν (z) Bessel function Second Kind H_ν (z) Struve H function
$$\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{{C}_{\mathrm{1}} {J}_{\nu} \left({t}\right)+{C}_{\mathrm{2}} {Y}_{\nu} \left({t}\right)+\boldsymbol{\mathrm{H}}_{\nu} \left({t}\right)}{{C}_{\mathrm{1}} {J}_{\nu} \left({t}\right)+{C}_{\mathrm{2}} {Y}_{\nu} \left({t}\right)}=?? \\ $$$$\nu\in\mathbb{R} \\ $$$${J}_{\nu} \left({z}\right)\:\mathrm{Bessel}\:\mathrm{function}\:\mathrm{First}\:\mathrm{kind} \\ $$$${Y}_{\nu} \left({z}\right)\:\mathrm{Bessel}\:\mathrm{function}\:\mathrm{Second}\:\mathrm{Kind} \\ $$$$\boldsymbol{\mathrm{H}}_{\nu} \left({z}\right)\:\mathrm{Struve}\:\mathrm{H}\:\mathrm{function} \\ $$
Answered by MrGaster last updated on 10/May/25
lim_(t→0) ((C_1 J_ν (t)+C_2 Y_ν (t)+H_ν (t))/(C_1 J_ν (t)+C_2 Y_ν (t)))=lim_(t→0) ((C_1 (t^ν /(2^ν Γ(ν+1)))+C_2 (−((Γ(ν)t^(−ν) )/2))+(t^(ν+1) /(2^ν Γ(ν+(3/2)))))/(C_1 (t^ν /(2^ν Γ(ν+1)))+C_2 (−((Γ(ν)t^(−ν) )/2)))) =lim_(t→0) ((C_1 −((C_2 Γ(ν)t^(−ν) )/2)+(t^(ν+1) /(2^ν Γ(ν+(3/2)))))/(C_1 t^ν −((C_2 Γ(ν)t^(−ν) )/2)))=lim_(t→0) (1+(t^(ν+1) /((2^ν Γ(ν+(3/2)))/(C_1 t^ν −((C_2 Γ(ν)t^(−ν) )/2)))))=1
$$\mathrm{lim}_{{t}\rightarrow\mathrm{0}} \:\:\frac{{C}_{\mathrm{1}} {J}_{\nu} \left({t}\right)+{C}_{\mathrm{2}} {Y}_{\nu} \left({t}\right)+\boldsymbol{\mathrm{H}}_{\nu} \left({t}\right)}{{C}_{\mathrm{1}} {J}_{\nu} \left({t}\right)+{C}_{\mathrm{2}} {Y}_{\nu} \left({t}\right)}=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{C}_{\mathrm{1}} \frac{{t}^{\nu} }{\mathrm{2}^{\nu} \Gamma\left(\nu+\mathrm{1}\right)}+{C}_{\mathrm{2}} \left(−\frac{\Gamma\left(\nu\right){t}^{−\nu} }{\mathrm{2}}\right)+\frac{{t}^{\nu+\mathrm{1}} }{\mathrm{2}^{\nu} \Gamma\left(\nu+\frac{\mathrm{3}}{\mathrm{2}}\right)}}{{C}_{\mathrm{1}} \frac{{t}^{\nu} }{\mathrm{2}^{\nu} \Gamma\left(\nu+\mathrm{1}\right)}+{C}_{\mathrm{2}} \left(−\frac{\Gamma\left(\nu\right){t}^{−\nu} }{\mathrm{2}}\right)} \\ $$$$=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{C}_{\mathrm{1}} −\frac{{C}_{\mathrm{2}} \Gamma\left(\nu\right){t}^{−\nu} }{\mathrm{2}}+\frac{{t}^{\nu+\mathrm{1}} }{\mathrm{2}^{\nu} \Gamma\left(\nu+\frac{\mathrm{3}}{\mathrm{2}}\right)}}{{C}_{\mathrm{1}} {t}^{\nu} −\frac{{C}_{\mathrm{2}} \Gamma\left(\nu\right){t}^{−\nu} }{\mathrm{2}}}=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{1}+\frac{{t}^{\nu+\mathrm{1}} }{\frac{\mathrm{2}^{\nu} \Gamma\left(\nu+\frac{\mathrm{3}}{\mathrm{2}}\right)}{{C}_{\mathrm{1}} {t}^{\nu} −\frac{{C}_{\mathrm{2}} \Gamma\left(\nu\right){t}^{−\nu} }{\mathrm{2}}}}\right)=\mathrm{1} \\ $$
Commented by SdC355 last updated on 10/May/25
Goood
$$\mathrm{Goood} \\ $$

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