prove-the-Following-Equation-J-z-and-Y-z-are-Bessel-function-J-1-2-z-1-1-Y-1-2-z-Y-1-2-z-1-J-1-2-z-Z-Do-Not-prove-using-the-equations-pres

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Question Number 212648 by issac last updated on 20/Oct/24

prove the Following Equation. J_ν (z) and Y_ν (z) are Bessel function J_(−ν−(1/2)) (z)=(−1)^(ν+1) Y_(ν+(1/2)) (z) Y_(−ν−(1/2)) (z)=(−1)^ν J_(ν+(1/2)) (z) ν∈Z^+ Do Not prove using the equations presented above

$$\mathrm{prove}\:\mathrm{the}\:\mathrm{Following}\:\mathrm{Equation}. \\ $$$$\:{J}_{\nu} \left({z}\right)\:\mathrm{and}\:{Y}_{\nu} \left({z}\right)\:\mathrm{are}\:\:\mathrm{Bessel}\:\mathrm{function} \\ $$$${J}_{−\nu−\frac{\mathrm{1}}{\mathrm{2}}} \left({z}\right)=\left(−\mathrm{1}\right)^{\nu+\mathrm{1}} {Y}_{\nu+\frac{\mathrm{1}}{\mathrm{2}}} \left({z}\right) \\ $$$${Y}_{−\nu−\frac{\mathrm{1}}{\mathrm{2}}} \left({z}\right)=\left(−\mathrm{1}\right)^{\nu} {J}_{\nu+\frac{\mathrm{1}}{\mathrm{2}}} \left({z}\right) \\ $$$$\nu\in\mathbb{Z}^{+} \\ $$$$\mathrm{Do}\:\mathrm{Not}\:\mathrm{prove}\:\mathrm{using}\:\mathrm{the}\:\mathrm{equations} \\ $$$$\mathrm{presented}\:\mathrm{above} \\ $$

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prove-the-Following-Equation-J-z-and-Y-z-are-Bessel-function-J-1-2-z-1-1-Y-1-2-z-Y-1-2-z-1-J-1-2-z-Z-Do-Not-prove-using-the-equations-pres

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