Die allgemeine Lösung einer zylindrischen Differentialgleichung vierter Ordnung nullten Parameterwertes
Applications of Mathematics, Tome 16 (1971) no. 3, pp. 203-214
The paper gives a comprehensive review of general solutions of the ordinary linear differential equation (1) $\Delta^2w+2\epsilon\Delta w+w=0, \ \Delta=d^2/d\rho^2+(1/\rho)(d/d\rho)$, the particular solution of which is represented by the Bessel function $w=Z_0(\rho\sqrt{\lambda})$of zero index with a real, imaginary or complex argument, respectively.
In the case $\epsilon = \pm1$ the corresponding characteristic equation $\lambda^2 - 2\epsilon\lambda + 1=0$ evidently zields one double root $\lambda=\pm 1$; then another independent particular solution of Eq. (1) is represented by the function $w=\rho Z_1(\rho\sqrt{\pm 1})$.
Generally it is proved that the general solution of a double Bessel equation of the $v$-th index (2) $[\Delta + \lambda - (v/\rho)^2]^2w=0$ can be written in the form $w=A_1J_v(\rho sqrt{\lambda})+A_2\rho J_{v+1}(\rho\sqrt{\lambda})+A_3Y_v(\rho\sqrt{\lambda})+A_4\rho Y_{v+1}(\rho\sqrt{\lambda})$ where $A_1$ to $A_4$ denote the constants of integration and $J_v(\rho \sqrt{\lambda}),\ Y_v(\rho \sqrt{\lambda})$ are the Bessel functions of the $v$-th index of the first and second kinds, respectively.
The paper gives a comprehensive review of general solutions of the ordinary linear differential equation (1) $\Delta^2w+2\epsilon\Delta w+w=0, \ \Delta=d^2/d\rho^2+(1/\rho)(d/d\rho)$, the particular solution of which is represented by the Bessel function $w=Z_0(\rho\sqrt{\lambda})$of zero index with a real, imaginary or complex argument, respectively.
In the case $\epsilon = \pm1$ the corresponding characteristic equation $\lambda^2 - 2\epsilon\lambda + 1=0$ evidently zields one double root $\lambda=\pm 1$; then another independent particular solution of Eq. (1) is represented by the function $w=\rho Z_1(\rho\sqrt{\pm 1})$.
Generally it is proved that the general solution of a double Bessel equation of the $v$-th index (2) $[\Delta + \lambda - (v/\rho)^2]^2w=0$ can be written in the form $w=A_1J_v(\rho sqrt{\lambda})+A_2\rho J_{v+1}(\rho\sqrt{\lambda})+A_3Y_v(\rho\sqrt{\lambda})+A_4\rho Y_{v+1}(\rho\sqrt{\lambda})$ where $A_1$ to $A_4$ denote the constants of integration and $J_v(\rho \sqrt{\lambda}),\ Y_v(\rho \sqrt{\lambda})$ are the Bessel functions of the $v$-th index of the first and second kinds, respectively.
@article{10_21136_AM_1971_103346,
author = {Panc, Vladim{\'\i}r},
title = {Die allgemeine {L\"osung} einer zylindrischen {Differentialgleichung} vierter {Ordnung} nullten {Parameterwertes}},
journal = {Applications of Mathematics},
pages = {203--214},
year = {1971},
volume = {16},
number = {3},
doi = {10.21136/AM.1971.103346},
zbl = {0224.34008},
language = {de},
url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1971.103346/}
}
TY - JOUR AU - Panc, Vladimír TI - Die allgemeine Lösung einer zylindrischen Differentialgleichung vierter Ordnung nullten Parameterwertes JO - Applications of Mathematics PY - 1971 SP - 203 EP - 214 VL - 16 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.21136/AM.1971.103346/ DO - 10.21136/AM.1971.103346 LA - de ID - 10_21136_AM_1971_103346 ER -
%0 Journal Article %A Panc, Vladimír %T Die allgemeine Lösung einer zylindrischen Differentialgleichung vierter Ordnung nullten Parameterwertes %J Applications of Mathematics %D 1971 %P 203-214 %V 16 %N 3 %U http://geodesic.mathdoc.fr/articles/10.21136/AM.1971.103346/ %R 10.21136/AM.1971.103346 %G de %F 10_21136_AM_1971_103346
Panc, Vladimír. Die allgemeine Lösung einer zylindrischen Differentialgleichung vierter Ordnung nullten Parameterwertes. Applications of Mathematics, Tome 16 (1971) no. 3, pp. 203-214. doi: 10.21136/AM.1971.103346
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