KelvinKei Notations Traditional name Kelvin function of the second kind Traditional notation kei Mathematica StandardForm notation KelvinKei Primary definition 03.5.0.000.0 kei kei 0 Specific values Values at fixed points 03.5.03.000.0 kei0 Values at infinities 03.5.03.000.0 lim keix 0 x 03.5.03.0003.0 lim keix x General characteristics Domain and analyticity kei is an analytical function of, which is defined over the whole complex -plane. 03.5.0.000.0 kei
http://functions.wolfram.com Symmetries and periodicities Mirror symmetry 03.5.0.000.0 kei kei ;, 0 Periodicity No periodicity Poles and essential singularities The function kei has an essential singularity at. At the same time, the point is a branch point. 03.5.0.0003.0 ing kei, Branch points The function kei has two branch points: 0,. At the same time, the point is an essential singularity. 03.5.0.000.0 kei 0, 03.5.0.0005.0 kei, 0 log 03.5.0.0006.0 kei, log Branch cuts The function kei is a single-valued function on the -plane cut along the interval, 0 where it is continuous from above. 03.5.0.0007.0 kei, 0, 03.5.0.0008.0 lim keix Ε keix ; x x 0 Ε0 03.5.0.0009.0 lim keix Ε keix beix ; x x 0 Ε0 Series representations Generalied power series Expansions at generic point 0
http://functions.wolfram.com 3 03.5.06.000.0 kei kei 0 arg 0 arg 0 arg 0 arg 0 bei 0 arg 0 bei 0 ber 0 kei 0 ker 0 arg 0 0 ber 0 ber 0 ker 0 ker 0 0 ; 0 03.5.06.000.0 kei k 0 0 k kei k k 0 03.5.06.0003.0 kei k 0 k G 3,7 3,3 0, ; arg 0 k, k, 3k k, k, k, 0,,, 3 0 k ; arg 0 03.5.06.000.0 kei k 0 k 3k k k j 0 k j k kei jk 0 k arg 0 arg 0 bei k j 0 k ker jk 0 k arg 0 arg 0 ber k j 0 k k j 0 j k kei jk 0 k arg 0 arg 0 bei jk 0 k ker jk 0 k arg 0 arg 0 ber jk 0 0 k 03.5.06.0005.0 kei kei 0 arg 0 Expansions on branch cuts arg 0 bei 0 O 0 kei 03.5.06.0006.0 arg x arg x beix keix argx bei x ber x kei x ker x x berx ber x kerx ker x x ; x x x 0
http://functions.wolfram.com 03.5.06.0007.0 kei k 0 k 3k k k j 0 k j k kei jk x k arg x bei k j x k ker jk x k arg x ber k j x k k j 0 j k kei jk x k arg x bei jk x k ker jk x k arg x ber jk x x k ; x x 0 kei 03.5.06.0008.0 k 0 k arg x, G x,6, k, 3k k, 0,,, 3, k G 3,3 x 3,7, k, k, 3k k, k, k, 0,,, 3 x k ; x x 0 03.5.06.0009.0 arg x kei keix beix O x ; x x 0 Expansions at 0 For the function itself 03.5.06.000.0 kei 8 6 7 56 log 8 576 3 686 00 6 356 37 60 8 ; 0 8 000 03.5.06.00.0 kei k k k 0 k k Ψ k k k 0 k log k k k 0 k 03.5.06.00.0 kei 0 F 3 ;,, ; 56 log 0F 3 ;, 3, 3 ; 56 k 0 k Ψ k k k 03.5.06.003.0 kei 8 I 0 J 0 I 0 J 0 log k 0 k Ψ k k k 03.5.06.00.0 kei O O log O ; 0
http://functions.wolfram.com 5 03.5.06.005.0 kei O log O log O For small integer powers of the function kei 03.5.06.006.0 3 log6 6 log log 3 8 log6 5 6 8 log log3 8 log log log 5 8 log 096 9 56 log log 3 log log log 3 536 8 3 log log log log 3 8 6 log log08 3 8 log096 6 8 log log096 6 log 8 80 log 7 680 05 log 0 log 7 838 log 80 log 0 log 80 7 8 6 8 6 3 000 3 7 0 800 log log 7 log log 630 8 ; 0 86 60 log 60 log 60 73
http://functions.wolfram.com 6 kei 03.5.06.007.0 6 k k 6 k 0 k 3 k 3 k k 3 log log Ψk 3 Ψk 5 6 Ψ k 3 k k 0 k 3 6 k k 3 k 0 k k3 Ψ k log6 log 3 Ψk Ψ k log 8 8 log 8 log 3 Ψk log8 8 log 3 Ψk 3 Ψ k Ψ k k k k 3 6 k k k 0 3 3 k k 8 log Ψk log Ψ k log Ψ k log Ψ k 3 log 3 9 Ψk 9 Ψ k Ψ k 3 Ψ k 3 Ψ k Ψk 3 Ψ k 3 Ψk 9 Ψk Ψ k 3 Ψ k Ψ k 3 Ψ k Ψ k 3 Ψ k Ψ k 3 3 Ψ k 3 Ψ k Ψ k Ψ k 3 03.5.06.008.0 kei 6 log O Asymptotic series expansions Expansions for any in exponential form Using exponential function with branch cut-free arguments
http://functions.wolfram.com 7 kei 03.5.06.009.0 8 3 3 3 3 3 log 3 log 3 3 log log O 3 8 3 3 3 log log 3 3 8 log log O 9 8 3 3 log log 3 3 3 log 3 log O 75 0 3 3 8 log log 3 3 3 log log 3 O ;
http://functions.wolfram.com 8 kei 03.5.06.000.0 8 3 3 n k k k 0 k k 3 3 3 k 3 3 3 3 3 log log k log 3 log 3 n k k k 0 k k 3 3 3 k 3 3 3 3 3 log log k log 3 log ; n
http://functions.wolfram.com 9 kei 03.5.06.00.0 8 3 3 3 3 3 log 3 log 3 3 log log 8 F 3 8, 8, 3 8, 3 8, 5 8, 5 8, 7 8, 7 8 ;,, 3 6 ; 3 8 3 3 3 log log 3 3 8 log log 3 8F 3 8, 3 8, 5 8, 5 8, 7 8, 7 8, 9 8, 9 8 ;, 3, 5 6 ; 9 8 3 3 log log 3 3 3 log 3 log 5 8F 3 8, 5 8, 7 8, 7 8, 9 8, 9 8, 8, 8 ; 3, 5, 3 6 75 ; 0 3 3 8 log log 3 3 3 log log 3 7 8F 3 8, 7 8, 9 8, 9 8, 8, 8, 3 8, 3 8 ; 5, 3, 7 6 ; ;
http://functions.wolfram.com 0 kei 03.5.06.00.0 8 3 3 3 3 3 log 3 log 3 3 log log O ; 03.5.06.003.0 kei 58 38 58 3 arg ; arg 3 True Residue representations 03.5.06.00.0 kei res s j 0 s s s s j j 0 res s s s s s j Integral representations On the real axis Contour integral representations Limit representations Generating functions Differential equations Ordinary linear differential equations and wronskians 03.5.3.000.0 w w 3 3 w w w 0 ; w c ber c bei c 3 ker c kei
http://functions.wolfram.com 03.5.3.000.0 W ber, bei, ker, kei 03.5.3.0003.0 g g 3 w g 3 g 3 g g g w 3 g g 6 g g g g g 3 g 5 g g g w g g 6 g g g g g 3 g 3 g 6 g g g g 0 g 3 g g 3 g 5 g 3 g 3 w g g 7 w 0 ; w c berg c beig c 3 kerg c keig 03.5.3.000.0 W berg, beig, kerg, keig g 6 03.5.3.0005.0 g g g 3 h w g 3 g h g 3 g g g g h h 3 w 3 g g g 6 g g g g g 3 g 5 g g h 6 g g h g g h g 3 g h g h g g h h w g g 6 g g g g g 3 g 3 g 6 g g g g 0 g 3 g g 3 g 5 g 3 g 3 h 3 g g h g 3 g h g 3 g g h 3 3 h g g g 9 g h h g 3 g 5 g h g h g g h h g g h g 3 g h g h g 3 g 3 h 3 hw g h g 7 g h 36 h h h 8 h h 3 h h 6 h h h g 3 g 3 h g 3 g g 6 h 3 6 h h h h h 3 g g h h h h g 6 g g g g g 3 g 5 g g g g h 3 h g 6 g g g g g 3 g 3 g 6 g g g g 0 g 3 g g 3 g 5 g 3 g 3 w 0 ; w c h berg c h beig c 3 h kerg c h keig 03.5.3.0006.0 W h berg, h beig, h kerg, h keig h g 6 03.5.3.0007.0 g w 6 r s 3 w 3 r s r 6 s s 7 w r s s s r s w a r r s r s 3 r s w 0 ; w c s bera r c s beia r c 3 s kera r c s keia r 03.5.3.0008.0 W s bera r, s beia r, s kera r, s keia r a r 6 r s6 03.5.3.0009.0 w logr logs w 3 log r 6 logs logr 3 log s w logr logs log s logr logs w a log r r log s logr log 3 s log r log s w 0 ; w c s bera r c s beia r c 3 s kera r c s keia r 03.5.3.000.0 W s bera r, s beia r, s kera r, s keia r a r s log 6 r
http://functions.wolfram.com Transformations Transformations and argument simplifications Argument involving basic arithmetic operations 03.5.6.000.0 kei kei bei log log 03.5.6.000.0 kei kei ber log logbei 03.5.6.0003.0 kei kei ber log logbei 03.5.6.000.0 kei kei ber bei log 3 log 03.5.6.0005.0 kei 3 kei bei log log 03.5.6.0006.0 kei 3 kei ber bei log 3 log 03.5.6.0007.0 kei kei bei log log ber Addition formulas 03.5.6.0008.0 kei ber k kei k bei k ker k ; k 03.5.6.0009.0 kei ber k kei k bei k ker k ; k Multiple arguments 03.5.6.000.0 k kei k k 0 k cos 3 k kei k ker k sin 3 k ; Related transformations Involving ker
http://functions.wolfram.com 3 03.5.6.00.0 kei ker J 0 log log Y 0 03.5.6.00.0 kei ker K 0 I 0 log log Differentiation Low-order differentiation 03.5.0.000.0 kei kei ker 03.5.0.000.0 kei ker ker Symbolic differentiation 03.5.0.0003.0 n kei 3 n n n n k 0 n k n kei kn n ker kn n n k 0 k n ker kn n kei kn ; n n kei n 03.5.0.000.0 3 n n n n k 0 k n k n k n kei kn n ker kn n k n kei kn n ker kn ; n 03.5.0.0005.0 n kei n G 3,3 3,7, n, n, 3n n, n, n, 0,,, 3 ; n Fractional integro-differentiation 03.5.0.0006.0 Α kei k k k log Ψ k Α k Α k 0 k k Α 3 Α k k Α log, k Α k k k k k k 0 k k 0 k k Α
http://functions.wolfram.com Α kei Α 03.5.0.0007.0 7 Α log Α F 5 Α 3 3 Α F 5, 3 ;, Α, Α, 3 Α k k k Ψ k Α k k 0 k k Α 3 3, 5 ; 3, 3 Α, Α, 5 Α, 3 Α, Α Α ; 56 k 0 ; 56 k k Α log, k k k Integration Indefinite integration 03.5..000.0 keia 6 G 3, a,5, 3 0,,,, 0 Definite integration 03.5..000.0 0 t Α p t keitt 3 Α3 p Α 3 Α p Α cos Α F 3 6 cos Α F 3 Α p Α cos Α sin Α F 3 Α F 3 Α 3, Α 3, Α 5, Α 5 ; 5, 3, 7 ; p, Α, Α 3, Α 3 ;, 3, 5 ; p Α, Α, Α, Α ; 3, 5, 3 ; p, Α, Α, Α ;,, 3 ; p ; ReΑ 0 Rep Integral transforms Laplace transforms 03.5..000.0 t keit 3F,, ; 3, 5 ; cos tan sin tan ; Re Mellin transforms
http://functions.wolfram.com 5 03.5..000.0 t keit sin ; Re 0 Representations through more general functions Through hypergeometric functions Involving hypergeometric U 03.5.6.000.0 kei 3 U,, 3 U,, 8 log log 0 F ; ; 8 log log3 0 F ; ; Through Meijer G Classical cases for the direct function itself 03.5.6.000.0 kei G 3,0 0, 56 Classical cases for powers of kei 0,,, 0 ; arg kei 03.5.6.0003.0 6,0 G 0, 6 0, 0, 0, 8 5,0 G,6 6, 3 0, 0, 0,,, Brychkov Yu.A. (006) 03.5.6.000.0 kei 6,0 G 0, 6 0, 0, 0, 8 5,0 G,6 6, 3 0, 0, 0,,, ; arg Brychkov Yu.A. (006) Classical cases involving bei bei 03.5.6.0005.0 kei 8,0 G 0, 6 0, 0, 0, 8 3, G,6 6, 3 0, 0,, 0,, Brychkov Yu.A. (006) 03.5.6.0006.0 bei kei 8,0 G 0, 6 0, 0, 0, 8 3, G,6 6, 3 0, 0,, 0,, ; 0 arg
http://functions.wolfram.com 6 Brychkov Yu.A. (006) Classical cases involving ber ber 03.5.6.0007.0 kei 8,0 G 0, 6 0,, 0, 0 8 3, G,6 6, 3 0,,, 0, 0, Brychkov Yu.A. (006) 03.5.6.0008.0 ber kei 8,0 G 0, 6 0,, 0, 0 8 3, G,6 6, 3 0,,, 0, 0, ; 0 arg Brychkov Yu.A. (006) Classical cases involving powers of ker kei 03.5.6.0009.0 ker 8 Brychkov Yu.A. (006),0 G 0, 6 0, 0, 0, kei 03.5.6.000.0 ker 5,0 G,6 6, 3 0, 0, 0,,, Brychkov Yu.A. (006) 03.5.6.00.0 kei ker 8,0 G 0, 6 0, 0, 0, ; arg Brychkov Yu.A. (006) 03.5.6.00.0 kei ker 5,0 G,6 6, 3 0, 0, 0,,, ; arg Brychkov Yu.A. (006) Classical cases involving ker kei 03.5.6.003.0 ker 8 5,0 G,6 6, 3 0, 0,,,, 0 Brychkov Yu.A. (006)
http://functions.wolfram.com 7 03.5.6.00.0 kei ker 8 G 5,0,6 6, 3 0, 0,,,, 0 ; arg Brychkov Yu.A. (006) Classical cases involving ber, bei and ker bei 03.5.6.005.0 kei ber Brychkov Yu.A. (006) ker,0 G 0, 6 0, 0, 0, bei 03.5.6.006.0 kei ber Brychkov Yu.A. (006) ker 5 3, G,6 6, 3 0, 0,, 0,, ber 03.5.6.007.0 kei bei Brychkov Yu.A. (006) ker 5 3, G,6 6, 3 0,,, 0, 0, bei 03.5.6.008.0 ker ber Brychkov Yu.A. (006) kei,0 G 0, 6 0,, 0, 0 03.5.6.009.0 bei kei ber ker,0 G 0, 6 0, 0, 0, ; arg Brychkov Yu.A. (006) 03.5.6.000.0 bei kei ber ker 5 3, G,6 6, 3 0, 0,, 0,, ; arg Brychkov Yu.A. (006) 03.5.6.00.0 ber kei bei ker 5 3, G,6 6, 3 0,,, 0, 0, ; arg Brychkov Yu.A. (006)
http://functions.wolfram.com 8 03.5.6.00.0 bei ker ber kei Brychkov Yu.A. (006),0 G 0, 6 0,, 0, 0 ; arg 3 3 arg arg Classical cases involving Bessel J 03.5.6.003.0 J 0 kei 8,0 G 0, 6 0, 0, 0,,0 G 0, 6 0,, 0, 0 3, G,6 6, 3 0,,, 0, 0, 3, G,6 6, 3 0, 0,, 0,, ; arg Classical cases involving Bessel I 03.5.6.00.0 I 0 kei 8,0 G 0, 6 0, 0, 0,,0 G 0, 6 0,, 0, 0 3, G,6 6, 3 0, 0,, 0,, 3, G,6 6, 3 0,,, 0, 0, ; arg Classical cases involving Bessel K K 0 03.5.6.005.0 kei 6,0 G 0,, 0, 0, 0, 8 6,0 G,6,, 3 0, 0, 0,,, ; arg 0 Classical cases involving 0 F 0F ; ; 03.5.6.006.0 kei 8,0 G 0, 6 0, 0, 0,,0 G 0, 6 0,, 0, 0 3, G,6 6, 3 0, 0,, 0,, 3, G,6 6, 3 0,,, 0, 0, 03.5.6.007.0 0F ; ; kei 8,0 G 0, 6 0, 0, 0,,0 G 0, 6 0,, 0, 0 3, G,6 6, 3 0, 0,, 0,, 3, G,6 6, 3 0,,, 0, 0, ; arg Generalied cases for the direct function itself
http://functions.wolfram.com 9 03.5.6.008.0 kei G 3,0 0,, 0,,, 0 Generalied cases for powers of kei 03.5.6.009.0 kei 6,0 G 0,, 0, 0, 0, G 5,0 7,6,, 3 0, 0, 0,,, Brychkov Yu.A. (006) Generalied cases involving bei 03.5.6.0030.0 bei kei 8,0 G 0,, 0, 0, 0, 8 3, G,6,, 3 0, 0,, 0,, Brychkov Yu.A. (006) Generalied cases involving ber 03.5.6.003.0 ber kei 8,0 G 0,, 0,, 0, 0 8 3, G,6,, 3 0,,, 0, 0, Brychkov Yu.A. (006) Generalied cases involving powers of ker 03.5.6.003.0 kei ker 8,0 G 0,, 0, 0, 0, Brychkov Yu.A. (006) 03.5.6.0033.0 kei ker G 5,0,6,, 3 0, 0, 0,,, Brychkov Yu.A. (006) Generalied cases involving ker 03.5.6.003.0 kei ker 8 G 5,0,6,, 3 0, 0,,,, 0 Brychkov Yu.A. (006) Generalied cases involving ber, bei and ker
http://functions.wolfram.com 0 03.5.6.0035.0 bei kei ber ker,0 G 0,, 0, 0, 0, Brychkov Yu.A. (006) 03.5.6.0036.0 bei kei ber ker Brychkov Yu.A. (006) 3, G,6,, 3 0, 0,, 0,, 03.5.6.0037.0 bei ker ber kei Brychkov Yu.A. (006) 3, G,6,, 3 0,,, 0, 0, 03.5.6.0038.0 bei ker ber kei,0 G 0,, 0,, 0, 0 Brychkov Yu.A. (006) Generalied cases involving Bessel J 03.5.6.0039.0 J 0 kei 8,0 G 0,, 0, 0, 0,,0 G 0,, 0,, 0, 0 3, G,6,, 3 0,,, 0, 0, 3, G,6,, 3 0, 0,, 0,, Generalied cases involving Bessel I 03.5.6.000.0 I 0 kei 8,0 G 0,, 0, 0, 0,,0 G 0,, 0,, 0, 0 3, G,6,, 3 0, 0,, 0,, 3, G,6,, 3 0,,, 0, 0, Generalied cases involving Bessel K K 0 03.5.6.00.0 kei 6,0 G 0,, 0, 0, 0, 8 6,0 G,6,, 3 0, 0, 0,,, ; arg 3
http://functions.wolfram.com Generalied cases involving 0 F 03.5.6.00.0 0F ; ; kei 8,0 G 0,, 0, 0, 0,,0 G 0,, 0,, 0, 0 3, G,6,, 3 0, 0,, 0,, 3, G,6,, 3 0,,, 0, 0, Representations through equivalent functions With related functions 03.5.7.000.0 kei K 0 Y 0 log log bei ber 03.5.7.000.0 kei 8 K 0 Y 0 log log I 0 log log J 0 kei 03.5.7.0003.0 I 0 3 J 0 K 0 K 0 J 0 Y 0 Y 0 3 arg True 03.5.7.000.0 kei ker log log J 0 Y 0 03.5.7.0005.0 kei ker 3 J 0 Y 0 Y 0 J 0 3 arg True 03.5.7.0006.0 kei ker I 0 log log K 0 03.5.7.0007.0 kei ker I 0 K 0 K 0 3 arg True Theorems History
http://functions.wolfram.com Copyright This document was downloaded from functions.wolfram.com, a comprehensive online compendium of formulas involving the special functions of mathematics. For a key to the notations used here, see http://functions.wolfram.com/notations/. Please cite this document by referring to the functions.wolfram.com page from which it was downloaded, for example: http://functions.wolfram.com/constants/e/ To refer to a particular formula, cite functions.wolfram.com followed by the citation number. e.g.: http://functions.wolfram.com/0.03.03.000.0 This document is currently in a preliminary form. If you have comments or suggestions, please email comments@functions.wolfram.com. 00-008, Wolfram Research, Inc.