Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation. Mathematica StandardForm notation

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1 KelvinKei Notations Traditional name Kelvin function of the second kind Traditional notation kei Mathematica StandardForm notation KelvinKei Primary definition kei kei 0 Specific values Values at fixed points kei0 Values at infinities lim keix 0 x lim keix x General characteristics Domain and analyticity kei is an analytical function of, which is defined over the whole complex -plane kei

2 Symmetries and periodicities Mirror symmetry 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 ing kei, Branch points The function kei has two branch points: 0,. At the same time, the point is an essential singularity kei 0, kei, 0 log 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 kei, 0, lim keix Ε keix ; x x 0 Ε lim keix Ε keix beix ; x x 0 Ε0 Series representations Generalied power series Expansions at generic point 0

3 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 ; kei k 0 0 k kei k k kei k 0 k G 3,7 3,3 0, ; arg 0 k, k, 3k k, k, k, 0,,, 3 0 k ; arg 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 kei kei 0 arg 0 Expansions on branch cuts arg 0 bei 0 O 0 kei 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

4 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 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 arg x kei keix beix O x ; x x 0 Expansions at 0 For the function itself kei log ; kei k k k 0 k k Ψ k k k 0 k log k k k 0 k kei 0 F 3 ;,, ; 56 log 0F 3 ;, 3, 3 ; 56 k 0 k Ψ k k k kei 8 I 0 J 0 I 0 J 0 log k 0 k Ψ k k k kei O O log O ; 0

5 kei O log O log O For small integer powers of the function kei log6 6 log log 3 8 log log log3 8 log log log 5 8 log log log 3 log log log log log log log log log log log log096 6 log 8 80 log log 0 log log 80 log 0 log log log 7 log log ; log 60 log 60 73

6 6 kei 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 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 kei 6 log O Asymptotic series expansions Expansions for any in exponential form Using exponential function with branch cut-free arguments

7 7 kei log 3 log 3 3 log log O log log log log O log log log 3 log O log log log log 3 O ;

8 8 kei n k k k 0 k k k log log k log 3 log 3 n k k k 0 k k k log log k log 3 log ; n

9 9 kei 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 ; log log log log 3 8F 3 8, 3 8, 5 8, 5 8, 7 8, 7 8, 9 8, 9 8 ;, 3, 5 6 ; log log log 3 log 5 8F 3 8, 5 8, 7 8, 7 8, 9 8, 9 8, 8, 8 ; 3, 5, ; log log log log 3 7 8F 3 8, 7 8, 9 8, 9 8, 8, 8, 3 8, 3 8 ; 5, 3, 7 6 ; ;

10 0 kei log 3 log 3 3 log log O ; kei arg ; arg 3 True Residue representations 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 w w 3 3 w w w 0 ; w c ber c bei c 3 ker c kei

11 W ber, bei, ker, kei 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 W berg, beig, kerg, keig g 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 W h berg, h beig, h kerg, h keig h g 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 W s bera r, s beia r, s kera r, s keia r a r 6 r s 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 W s bera r, s beia r, s kera r, s keia r a r s log 6 r

12 Transformations Transformations and argument simplifications Argument involving basic arithmetic operations kei kei bei log log kei kei ber log logbei kei kei ber log logbei kei kei ber bei log 3 log kei 3 kei bei log log kei 3 kei ber bei log 3 log kei kei bei log log ber Addition formulas kei ber k kei k bei k ker k ; k kei ber k kei k bei k ker k ; k Multiple arguments k kei k k 0 k cos 3 k kei k ker k sin 3 k ; Related transformations Involving ker

13 kei ker J 0 log log Y kei ker K 0 I 0 log log Differentiation Low-order differentiation kei kei ker kei ker ker Symbolic differentiation 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 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 n kei n G 3,3 3,7, n, n, 3n n, n, n, 0,,, 3 ; n Fractional integro-differentiation 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

14 Α kei Α 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 keia 6 G 3, a,5, 3 0,,,, 0 Definite integration 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 t keit 3F,, ; 3, 5 ; cos tan sin tan ; Re Mellin transforms

15 t keit sin ; Re 0 Representations through more general functions Through hypergeometric functions Involving hypergeometric U 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 kei G 3,0 0, 56 Classical cases for powers of kei 0,,, 0 ; arg kei ,0 G 0, 6 0, 0, 0, 8 5,0 G,6 6, 3 0, 0, 0,,, Brychkov Yu.A. (006) 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 kei 8,0 G 0, 6 0, 0, 0, 8 3, G,6 6, 3 0, 0,, 0,, Brychkov Yu.A. (006) bei kei 8,0 G 0, 6 0, 0, 0, 8 3, G,6 6, 3 0, 0,, 0,, ; 0 arg

16 6 Brychkov Yu.A. (006) Classical cases involving ber ber kei 8,0 G 0, 6 0,, 0, 0 8 3, G,6 6, 3 0,,, 0, 0, Brychkov Yu.A. (006) 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 ker 8 Brychkov Yu.A. (006),0 G 0, 6 0, 0, 0, kei ker 5,0 G,6 6, 3 0, 0, 0,,, Brychkov Yu.A. (006) kei ker 8,0 G 0, 6 0, 0, 0, ; arg Brychkov Yu.A. (006) kei ker 5,0 G,6 6, 3 0, 0, 0,,, ; arg Brychkov Yu.A. (006) Classical cases involving ker kei ker 8 5,0 G,6 6, 3 0, 0,,,, 0 Brychkov Yu.A. (006)

17 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 kei ber Brychkov Yu.A. (006) ker,0 G 0, 6 0, 0, 0, bei kei ber Brychkov Yu.A. (006) ker 5 3, G,6 6, 3 0, 0,, 0,, ber kei bei Brychkov Yu.A. (006) ker 5 3, G,6 6, 3 0,,, 0, 0, bei ker ber Brychkov Yu.A. (006) kei,0 G 0, 6 0,, 0, bei kei ber ker,0 G 0, 6 0, 0, 0, ; arg Brychkov Yu.A. (006) bei kei ber ker 5 3, G,6 6, 3 0, 0,, 0,, ; arg Brychkov Yu.A. (006) ber kei bei ker 5 3, G,6 6, 3 0,,, 0, 0, ; arg Brychkov Yu.A. (006)

18 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 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 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 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 ; ; 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, F ; ; 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

19 kei G 3,0 0,, 0,,, 0 Generalied cases for powers of kei 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 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 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 kei ker 8,0 G 0,, 0, 0, 0, Brychkov Yu.A. (006) kei ker G 5,0,6,, 3 0, 0, 0,,, Brychkov Yu.A. (006) Generalied cases involving ker kei ker 8 G 5,0,6,, 3 0, 0,,,, 0 Brychkov Yu.A. (006) Generalied cases involving ber, bei and ker

20 bei kei ber ker,0 G 0,, 0, 0, 0, Brychkov Yu.A. (006) bei kei ber ker Brychkov Yu.A. (006) 3, G,6,, 3 0, 0,, 0,, bei ker ber kei Brychkov Yu.A. (006) 3, G,6,, 3 0,,, 0, 0, bei ker ber kei,0 G 0,, 0,, 0, 0 Brychkov Yu.A. (006) Generalied cases involving Bessel J 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 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 kei 6,0 G 0,, 0, 0, 0, 8 6,0 G,6,, 3 0, 0, 0,,, ; arg 3

21 Generalied cases involving 0 F F ; ; 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 kei K 0 Y 0 log log bei ber kei 8 K 0 Y 0 log log I 0 log log J 0 kei I 0 3 J 0 K 0 K 0 J 0 Y 0 Y 0 3 arg True kei ker log log J 0 Y kei ker 3 J 0 Y 0 Y 0 J 0 3 arg True kei ker I 0 log log K kei ker I 0 K 0 K 0 3 arg True Theorems History

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