Table 1 Inverse Sumudu transform of elementary functions

1

\(\frac{1}{x+a}\)

\(\frac{1}{a}e^{-\frac{w}{a}}\)

2

\(\frac{1}{x-a}\)

\(-\frac{1}{a}e^{\frac{w}{a}}\)

3

\(\frac{1}{(x+a)^{n}}\)

\(\frac{e^{-\frac{w}{2a}}}{a^{n}w} [ a(n+1) M_{n+1,\frac{1}{2}}( \frac{w}{a}) -(an-w)M_{n,\frac{1}{2}}( \frac{w}{a}) ]\)

4

\(\frac{x^{n}}{x+a}\)

\(\frac{(-1)^{-n}a^{n-1}e^{-\frac{w}{a}}}{n!} [ \Gamma ( n+1) -n\Gamma ( n,-\frac{w}{a}) ] \)

5

\(\frac{Ax+Ba}{x^{2}-a^{2}}\)

\(\frac{1}{2a} [ e^{-\frac{w}{a}}(A-B)-e^{\frac{w}{a}}(A+B) ] \)

6

\(\frac{Ax+Ba}{x^{2}+a^{2}}\)

\(\frac{1}{2a} [ e^{-\frac{iw}{a}}(iA+B)-e^{\frac{iw}{a}}(iA-B) ] \)

7

\(\frac{1}{\sqrt{x+a}}\)

\(\frac{1}{\sqrt{a}}e^{-\frac{w}{2a}}\mbox{I}_{0}( \frac{w}{2a}) \)

8

\(\frac{1}{(x+a)^{\frac{3}{2}}}\)

\(\frac{e^{-\frac{w}{2a}}}{a^{\frac{5}{2}}} [ (a-w)\mbox{I}_{0}( \frac{w}{2a}) +w\mbox{I}_{1}( \frac{w}{2a}) ]\)

9

\(\frac{\sqrt{x}}{x+a}\)

\(-\frac{e^{-\frac{w}{a}}}{\sqrt{a}}\mbox{erf}( \frac{i\sqrt{w}}{\sqrt{a}}) \)

10

\(\frac{\sqrt{x-b}}{x}\)

\(-\frac{ie^{\frac{w}{2b}}}{2\sqrt{b}} [ \mbox{I}_{0}( \frac{w}{2b}) - \mbox{I}_{1}( \frac{w}{2b}) ]\)

11

\(\frac{1}{\sqrt{x}(x+a)}\)

\(\frac{e^{-\frac{w}{a}}}{a^{2}\sqrt{w\pi }} [ ae^{\frac{w}{a}}+i\sqrt{aw\pi }\mbox{erf}( \frac{i\sqrt{w}}{\sqrt{a}}) ]\)

12

\(\frac{1}{\sqrt{x(x+a)}}\)

\(\frac{e^{\frac{w}{a}}}{\sqrt{aw\pi }}\)

13

\(\frac{1}{x\sqrt{x-b}}\)

\(-\frac{ie^{\frac{w}{2b}}}{2b^{\frac{3}{2}}} [ \mbox{I}_{0}( \frac{w}{2b}) +\mbox{I}_{1}( \frac{w}{2b}) ]\)

14

\(\frac{x}{\sqrt{x^{2}+a^{2}}}\)

\(-\frac{w\operatorname{csgn}(a)}{2a} [ \mbox{J}_{0}( \frac{w}{a})( \pi \textbf{H}_{1}( \frac{w}{a}) -2) - \pi \mbox{J}_{1}( \frac{w}{a}) \textbf{H}_{0}( \frac{w}{a}) ] \)

15

\(\frac{x}{\sqrt{b^{2}-x^{2}}}\)

\(\frac{w\operatorname{csgn}(b)}{2b} [ \mbox{I}_{0}( \frac{w}{b}) ( \pi \textbf{L}_{1}( \frac{w}{b}) +2) -\pi \mbox{I}_{1}( \frac{w}{b}) \textbf{L}_{0}( \frac{w}{b}) ] \)

16

\(\frac{x}{\sqrt{x^{2}-b^{2}}}\)

\(\frac{iw}{2b} [ \mbox{I}_{0}( \frac{w}{b}) ( \pi \textbf{L}_{1}( \frac{w}{b}) +2) -\pi \mbox{I} _{1}( \frac{w}{b}) \textbf{L}_{0}( \frac{w}{b}) ] \)

17

\(\frac{1}{x+\sqrt{x^{2}+a^{2}}}\)

\(\frac{1}{2a^{3}}(2\operatorname{csgn}(a)\mbox{J}_{0}( \frac{w}{a}) ( a^{2}+w^{2}-\frac{\pi w^{2}}{2}\textbf{H}_{1}( \frac{w}{a}) )-2w( \operatorname{csgn}(a)\mbox{J}_{1}( \frac{w}{a}) +a ( a-\frac{\pi w}{2} \textbf{H}_{0}( \frac{w}{a}) ) )) \)

18

\(( x+\sqrt{1+x^{2}}) ^{n}+( x-\sqrt{1+x^{2}})^{n} \)

\((e^{in\pi }+1){}_{2}F_{3}( \frac{n}{2},\frac{n}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w^{2}}{4}) +nw(e^{in\pi }-1){}_{2}F_{3} ( \frac{1+n}{2},\frac{1-n}{2};1,\frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4})\)

19

\(\frac{( x+\sqrt{1+x^{2}}) ^{n}}{\sqrt{1+x^{2}}}\)

\({}_{2}F_{3}( \frac{1-n}{2},\frac{1+n}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w^{2}}{4}) +nw{}_{2}F_{3}( 1-\frac{n}{2},1+\frac{n}{2};1, \frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \)

20

\(\frac{( x-\sqrt{1+x^{2}}) ^{n}}{\sqrt{1+x^{2}}}\)

\((e^{in\pi })_{2}F_{3}( \frac{1-n}{2},\frac{1+n}{2};\frac{1}{2}, \frac{1}{2},1;-\frac{w^{2}}{4}) -nw{}_{2}F_{3}( 1-\frac{n}{2},1+\frac{n}{2};1,\frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \)

21

\(\frac{(x-b)^{v}}{x}\)

\((-1)^{v}b^{v-1} [ \mbox{L}_{v}( \frac{w}{b}) -\mbox{L} _{v}^{1}( \frac{w}{b}) ] \)

22

\(\frac{x^{v-1}}{1+x^{2}}\)

\(\frac{\sqrt{w}\mbox{S1}_{\frac{3}{2},\frac{1}{2}}(w)+v(v+1)w^{v-1}-w ^{v+1}}{(v+1)!} \)

23

\((1+x^{2})^{v-\frac{1}{2}}\)

\({}_{1}F_{2}( \frac{1}{2}-v;\frac{1}{2},1;-\frac{w^{2}}{4}) \)

24

\((x^{2}-b^{2})^{v-\frac{1}{2}}\)

\((-b)^{v-\frac{1}{2}}{}_{1}F_{2}( \frac{1}{2}-v;\frac{1}{2},1;\frac{w ^{2}}{4b^{2}})\)

25

\((b^{2}-x^{2})^{v-\frac{1}{2}}\)

\(\frac{2(b^{2})^{v-\frac{1}{2}}}{b^{2}(2v+1)} [ w^{2}\mbox{L}_{v+ \frac{1}{2}}^{1}( \frac{w^{2}}{b^{2}}) +( ( v+ \frac{1}{2}) b^{2}-w^{2}) \mbox{L}_{v+\frac{1}{2}}( \frac{w ^{2}}{b^{2}}) ]\)

26

\(x^{v-1}(x+a)^{\frac{1}{2}-v}\)

\(\frac{a^{\frac{1}{2}-v}w^{v-1}e^{-\frac{w}{2a}} ( \frac{w}{a}) ^{-\frac{v}{2}}}{av(v-1)!(v+1)w} [ a(av+w)( v+\frac{3}{2})\mbox{M}_{\frac{v+3}{2},\frac{v+1}{2}}( \frac{w}{a}) +( \frac{2w^{2}-2wa-va^{2}}{2}) \mbox{M}_{\frac{v+1}{2}, \frac{v+1}{2}}( \frac{w}{a}) ] \)

27

\(x^{v-1}(x+a)^{-\frac{1}{2}-v}\)

\(\frac{e^{-\frac{w}{2a}}( \frac{w}{a}) ^{-\frac{v}{2}} a^{- ( v+\frac{1}{2}) }w^{v-1}}{vw(v+1)(v-1)!} [ -( \frac{w+av}{2}) \mbox{M}_{\frac{v+1}{2},\frac{v+1}{2}}( \frac{w}{a}) + ( v+\frac{3}{2}) av\mbox{M}_{\frac{v+3}{2},\frac{v+1}{2}} ( \frac{w}{a}) ] \)

28

\(( \sqrt{x^{2}+1}+x) ^{v}\)

\({}_{2}F_{3}( \frac{v}{2},-\frac{v}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w^{2}}{4}) +vw{}_{2}F_{3}( \frac{1+v}{2},\frac{1-v}{2};1, \frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \)

29

\(( \sqrt{x^{2}+1}-x) ^{v}\)

\({}_{2}F_{3}( \frac{v}{2},-\frac{v}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w ^{2}}{4}) -vw{}_{2}F_{3}( \frac{1+v}{2},\frac{1-v}{2};1, \frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \)

30

\(\frac{( \sqrt{x^{2}+1}+x) ^{v}}{\sqrt{x^{2}+1}}\)

\({}_{2}F_{3}( \frac{1-v}{2},\frac{1+v}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w ^{2}}{4}) +vw{}_{2}F_{3}( 1-\frac{v}{2},1+\frac{v}{2};1, \frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \)

31

\(\frac{( \sqrt{x^{2}+1}-x) ^{v}}{\sqrt{x^{2}+1}}\)

\({}_{2}F_{3}( \frac{1-v}{2},\frac{1+v}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w ^{2}}{4}) -vw{}_{2}F_{3}( 1-\frac{v}{2},1+\frac{v}{2};1, \frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \)

32

\(\frac{( \sqrt{x^{2}-1}+x) ^{v}+( \sqrt{x^{2}-1}+x)^{-v}}{\sqrt{x^{2}-1}}\)

\(-\frac{(e^{i\pi v}+1)}{i^{v-1}}{}_{2}F_{3}( \frac{1-v}{2},\frac{1+v}{2};\frac{1}{2},\frac{1}{2},1;\frac{w^{2}}{4}) -\frac{(e^{i\pi v}-1)vw}{i^{v}}{}_{2}F_{3}( 1-\frac{v}{2},1+\frac{v}{2};1,\frac{3}{2},\frac{3}{2};\frac{w^{2}}{4})\)

33

\(( \sqrt{x+2a}+\sqrt{x}) ^{2v} -( \sqrt{x+2a}-\sqrt{x}) ^{2v}\)

\(\frac{4v\sqrt{w}2^{v+\frac{1}{2}}a^{v-\frac{1}{2}}}{\sqrt{\pi }}{}_{2}F_{2}( \frac{1}{2}+v,\frac{1}{2}-v;\frac{3}{2},\frac{3}{2};-\frac{w}{2a})\)

34

\(( \sqrt{x+b}+\sqrt{x-b}) ^{2v} -( \sqrt{x+b}-\sqrt{x-b}) ^{2v}\)

\(-ivwb^{v-1}( (1+i)^{2v}+(1-i)^{2v}) {}_{2}F_{3}( \frac{1+v}{2}, \frac{1-v}{2};1,\frac{3}{2},\frac{3}{2};\frac{w^{2}}{4b^{2}}) -b^{v}( -(1+i)^{2v}+(1-i)^{2v}) {}_{2}F_{3}( \frac{v}{2},- \frac{v}{2};\frac{1}{2},\frac{1}{2},1;,\frac{w^{2}}{4b^{2}} )\)

35

\(\frac{(2a)^{2v}( x+\sqrt{x^{2}+4a^{2}}) ^{2v}}{\sqrt{x^{3}+4a^{2}x}}\)

\(\frac{1}{24}( \operatorname{csgn}^{2v+1}(a)w^{\frac{3}{2}}a^{4v-3}16^{v}( v^{2}-\frac{1}{4}) ) \times {}_{2}F_{5}( \frac{3}{2}-v, \frac{3}{2}+v;\frac{5}{4},\frac{3}{2},\frac{3}{2},\frac{7}{4},2;-\frac{w^{2}}{1024a^{2}}) +\frac{1}{2}( \operatorname{csgn}^{2v}(a)va^{4v+2}16^{v}\sqrt{w}) \times {}_{2}F_{5}( 1+v,1-v; \frac{1}{2},\frac{3}{4},1,\frac{5}{4},\frac{3}{2};-\frac{w^{2}}{1024a^{2}})\)

36

\(\frac{( x+\sqrt{x^{2}-1}) ^{2v}+( x-\sqrt{x^{2}-1})^{2v}}{\sqrt{x}\sqrt{x^{2}-1}}\)

\(\frac{8iv\sqrt{w}}{\sqrt{\pi }\sin (\pi v)}( \cos^{2}(\pi v)-1){}_{2}F_{3}( 1+v,1-v;\frac{3}{4},\frac{5}{4},\frac{3}{2}; \frac{w^{2}}{4}) -\frac{2i\cos (\pi v)}{\sqrt{\pi w}}{}_{2}F_{3}( \frac{1}{2}+v,\frac{1}{2}-v;\frac{1}{4},\frac{1}{1},\frac{3}{4};\frac{w ^{2}}{4})\)

37

\(e^{-ax}\)

\(\mbox{J}_{0}(2\sqrt{aw})\)

38

\(xe^{-ax}\)

\(\frac{w\mbox{J}_{1}(2\sqrt{aw})}{\sqrt{aw}}\)

39

\(x^{v-1}e^{-ax}\)

\(\frac{w^{v-1}\Gamma (v)\mbox{J}_{v-1}(2\sqrt{aw})}{(v-1)!(aw)^{\frac{v-1}{2}}}\)

40

\(\frac{e^{-ax}-e^{-bx}}{x}\)

\(\frac{b\sqrt{a}\mbox{J}_{1}(2\sqrt{bw})-a\sqrt{b}\mbox{J}_{1}(2\sqrt{aw})}{\sqrt{abw}} \)

41

\(\frac{(1-e^{-ax})^{2}}{x^{2}}\)

\(\frac{-2a(\mbox{J}_{2}(2\sqrt{aw})-\mbox{J}_{2}(2\sqrt{2aw}))}{w}\)

42

\(\frac{1}{x}-\frac{(x+2)(1-e^{-x})}{2x^{2}}\)

\(\frac{w\mbox{I}_{3}(2\sqrt{-w})}{2(-w)^{\frac{3}{2}}}\)

43

\(e^{-\frac{x^{2}}{4a}}\)

\({}_{0}F_{2}( ;\frac{1}{2},1;-\frac{w^{2}}{16a})\)

44

\(xe^{-\frac{x^{2}}{4a}}\)

\(w{}_{0}F_{2}( ;1,\frac{3}{2};-\frac{w^{2}}{16a}) \)

45

\(\frac{e^{-\frac{x^{2}}{4a}}}{\sqrt{x}}\)

\(\frac{1}{\sqrt{\pi w}}{}_{0}F_{2}( ;\frac{1}{4},\frac{3}{4};-\frac{w ^{2}}{16a}) \)

46

\(x^{v-1}e^{-\frac{x^{2}}{8a}}\)

\(\frac{w^{v-1}}{(v-1)!}{}_{0}F_{2}( ;\frac{v}{2},\frac{v+1}{2};-\frac{w ^{2}}{32a}) \)

47

\(e^{-\frac{x}{4a}}\)

\(\mbox{I}_{0}( \sqrt{-\frac{w}{a}}) \)

48

\(\sqrt{x}e^{-\frac{x}{4a}}\)

\(2\sqrt{\frac{a}{\pi }}\sin ( \sqrt{\frac{w}{a}}) \)

49

\(\frac{e^{-\frac{x}{4a}}}{\sqrt{x}}\)

\(\frac{\cos ( \sqrt{\frac{w}{a}}) }{\sqrt{\pi w}} \)

50

\(\frac{e^{-\frac{x}{4a}}}{x^{\frac{3}{2}}}\)

\(-\frac{( \sqrt{aw}\sin ( \sqrt{\frac{w}{a}}) +a \cos ( \sqrt{\frac{w}{a}}) ) }{2a\sqrt{\pi }w^{ \frac{3}{2}}} \)

51

\(x^{v-1}e^{-\frac{x}{4a}}\)

\(\frac{(2w)^{v-1}\Gamma (v)( -\frac{w}{a}) ^{\frac{1-v}{2}} \mbox{I}_{v-1}( \sqrt{-\frac{w}{a}}) }{(v-1)!} \)

52

\(\frac{(e^{-\frac{x}{4a}}-1)}{\sqrt{x}}\)

\(\frac{1}{\sqrt{w\pi }} [ \cos ( \sqrt{\frac{w}{a}}) -1 ] \)

53

\(e^{-2\sqrt{a}\sqrt{x}}\)

\(-\frac{4\sqrt{aw}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{3}{2},\frac{3}{2};,aw) + {}_{0}F_{2}( ;\frac{1}{2},1;,aw) \)

54

\(\sqrt{x}e^{-2\sqrt{a}\sqrt{x}}\)

\(\frac{2\sqrt{w}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{1}{2},\frac{3}{2};,aw) - 2w\sqrt{a}_{0}F_{2}( ;\frac{3}{2},2;,aw) \)

55

\(\frac{e^{-2\sqrt{a}\sqrt{x}}}{\sqrt{x}}\)

\(\frac{1}{\sqrt{w\pi }} {}_{0}F_{2}( ;\frac{1}{2},\frac{1}{2};,aw) - 2\sqrt{a}_{0}F_{2}( ;1,\frac{3}{2};,aw) \)

56

\(\frac{e^{-2\sqrt{a}\sqrt{x}}}{\sqrt{2x}}\)

\(\frac{1}{\sqrt{2w\pi }} {}_{0}F_{2}( ;\frac{1}{2},\frac{1}{2};,aw) -\sqrt{2a}_{0}F_{2}( ;1,\frac{3}{2};,aw) \)

57

log(1 + ax)

Ei₁(aw)+ln(aw)+γ

58

log(x + a)

\(\mbox{Ei}_{1}( \frac{w}{a}) +\ln (w)+\gamma \)

59

\(\log (x^{2}-a^{2})\)

\(2 [ \ln (w)+\gamma -\mbox{Chi}( \frac{w}{a}) ] +i \pi \)

60

\(\log (x^{2}+a^{2})\)

\(2 [ \ln (w)+\gamma -\mbox{Ci}( \frac{w}{a}) ] \)

61

\(\frac{\log (x^{2}+a^{2})-\log (a^{2})}{x}\)

\(\frac{2}{w} [ 1-\cos ( \frac{w}{a}) ] \)

62

\(\log ( \frac{\sqrt{x+b}+\sqrt{x-b}}{\sqrt{2}\sqrt{b}})\)

\(\frac{1}{2b} [ -iw {}_{2}F_{3}( \frac{1}{2},\frac{1}{2};1, \frac{3}{2},\frac{3}{2};\frac{w^{2}}{4b^{2}}) - b( \ln (2)+2 \ln (i+1)) ] \)

63

\(\log ( \frac{\sqrt{x}+\sqrt{x+2a}}{\sqrt{2}\sqrt{a}})\)

\(\sqrt{\frac{2w}{a\pi }}{}_{2}F_{2}( \frac{1}{2},\frac{1}{2}; \frac{3}{2},\frac{3}{2};-\frac{w}{2a}) \)

64

\(\log ( \frac{\sqrt{x+ib}+\sqrt{x-ib}}{\sqrt{2}\sqrt{b}})\)

\(\frac{1}{2b} [ w{}_{2}F_{3}( \frac{1}{2},\frac{1}{2};1, \frac{3}{2},\frac{3}{2}; -\frac{w^{2}}{4b^{2}}+2i\pi b) ] \)

65

sin(ax)

\(\operatorname{bei}_{0}(2\sqrt{aw}) \)

66

xsin(ax)

\(-\frac{\sqrt{aw}}{2} [ \mbox{ber}_{1}(2\sqrt{w}) +\operatorname{bei} _{1}(2\sqrt{w}) ] \)

67

\(x^{n}\sin (ax)\)

\(\frac{aw^{n+1}}{(n+1)!}{}_{0}F_{3}( ;\frac{3}{2},\frac{n}{2}+1, \frac{n+3}{2};-\frac{(aw)^{2}}{16}) \)

68

\(x^{v-1}\sin (ax)\)

\(\frac{aw^{v}}{v!}{}_{0}F_{3}( ;\frac{3}{2},\frac{v}{2}+1, \frac{v+1}{2};-\frac{(aw)^{2}}{16}) \)

69

\(\frac{\sin (ax)}{x}\)

\(-\sqrt{\frac{a}{2w}} [ \mbox{ber}_{1}(2\sqrt{aw})-\operatorname{bei} _{1}(2\sqrt{aw}) ] \)

70

\(\frac{\sin^{2}(ax)}{x}\)

\(-\sqrt{\frac{a}{4w}} [ \mbox{ber}_{1}(2\sqrt{2aw})+\operatorname{bei} _{1}(2\sqrt{2aw}) ] \)

71

\(\frac{\sin^{3}(ax)}{x}\)

\(-\frac{3\sqrt{a}}{8\sqrt{3iw}} [ -\sqrt{3}( \mbox{I} _{1}( 2(-1)^{\frac{1}{4}}\sqrt{aw}) +\mbox{J}_{1}( 2(-1)^{ \frac{1}{4}}\sqrt{aw}) ) +\mbox{I}_{1}(2\sqrt{3iaw})+ \mbox{J}_{1}(2\sqrt{3iaw}) ] \)

72

\(\frac{\sin^{2}(ax)}{x^{2}}\)

\(\frac{\sqrt{a}}{2w^{\frac{3}{2}}} [ \mbox{ber}_{1}(2 \sqrt{2aw})+\operatorname{bei}_{1}(2\sqrt{2aw})+2\sqrt{aw}\operatorname{bei}_{0}(2 \sqrt{2aw}) ] \)

73

\(\sin (x^{2})\)

\(\frac{w^{2}}{2}{}_{0}F_{5}( ;\frac{3}{4},1,\frac{5}{4}, \frac{3}{2}, \frac{3}{2};-\frac{w^{2}}{1024}) \)

74

\(\frac{\sin (x^{2})}{x}\)

\(w{}_{0}F_{5}( ;\frac{1}{2},\frac{3}{4},1,\frac{5}{4}, \frac{3}{2};-\frac{w ^{2}}{1024})\)

75

\(\frac{\sin (x^{2})}{x^{2}}\)

\(w{}_{0}F_{5}( ;\frac{1}{4},\frac{1}{2},\frac{3}{2},1,\frac{3}{2} ;-\frac{w ^{2}}{1024}) \)

76

\(\sin (2\sqrt{a}\sqrt{x})\)

\(\frac{4\sqrt{aw}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{3}{2}, \frac{3}{2};-aw) \)

77

\(x^{n}\sin (2\sqrt{a}\sqrt{x})\)

\(\frac{2\sqrt{a}w^{n+\frac{1}{2}}}{( n+\frac{1}{2}) !} {}_{0}F_{2}( ;\frac{3}{2},\frac{3}{2}+n;-aw) \)

78

\(\frac{\sin (2\sqrt{a}\sqrt{x})}{x}\)

\(\frac{2\sqrt{a}}{\sqrt{w\pi }}{}_{0}F_{2}( ;\frac{1}{2}, \frac{3}{2};-aw) \)

79

\(\sqrt{x}\sin (2\sqrt{a}\sqrt{x})\)

\(2w\sqrt{a}{}_{0}F_{2}( ;\frac{3}{2},2;-aw) \)

80

\(\frac{\sin (2\sqrt{a}\sqrt{x})}{\sqrt{x}}\)

\(2\sqrt{a}{}_{0}F_{2}( ;1,\frac{3}{2};-aw) \)

81

\(x^{v-1}\sin (2\sqrt{a}\sqrt{x})\)

\(\frac{2\sqrt{a}w^{v-\frac{1}{2}}}{( v-\frac{1}{2}) !}{}_{0}F_{2}( ;\frac{3}{2},v+\frac{1}{2};-aw) \)

82

cos(ax)

\(\mbox{ber}_{0}(2\sqrt{aw}) \)

83

cos²(ax)

\(1-\frac{(aw)^{2}}{2}{}_{1}F_{4}( 1;\frac{3}{2},\frac{3}{2},2,2;- \frac{(aw)^{2}}{4}) \)

84

cos³(ax)

\(\frac{3}{8} [ \mbox{I}_{0}( 2(-1)^{\frac{1}{4}}\sqrt{aw}) + \mbox{J}_{0}( 2(-1)^{\frac{1}{4}}\sqrt{aw}) ] + \frac{1}{8} [ \mbox{I}_{0}( 2\sqrt{3iaw}) + \mbox{J} _{0}( 2\sqrt{3iaw}) ] \)

85

xcos(x)

\(-\sqrt{\frac{w}{2}} [ \mbox{ber}_{1}(2\sqrt{w})-\operatorname{bei}_{1}(2 \sqrt{w}) ] \)

86

\(x^{n}\cos (ax)\)

\(\frac{w^{n}}{n!}{}_{0}F_{3}( ;\frac{1}{2},\frac{n}{2}+1, \frac{n+1}{2};-\frac{(aw)^{2}}{16}) \)

87

\(x^{v-1}\cos (ax)\)

\(\frac{w^{v-1}}{(v-1)!}{}_{0}F_{3}( ;\frac{1}{2},\frac{v}{2}, \frac{v+1}{2};-\frac{(aw)^{2}}{16}) \)

88

\(\frac{1-\cos (ax)}{x}\)

\(-\sqrt{\frac{a}{2w}} [ \mbox{ber}_{1}(2\sqrt{aw})+\operatorname{bei} _{1}(2\sqrt{aw}) ] \)

89

\(\cos (x^{2}) \)

\({}_{0}F_{5}( ;\frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4},1;-\frac{w ^{2}}{1024}) \)

90

\(\cos (2\sqrt{a}\sqrt{x})\)

\({}_{0}F_{2}( ;\frac{1}{2},1;-aw) \)

91

\(x\sqrt{x}\cos (2\sqrt{a}\sqrt{x})\)

\(\frac{2\sqrt{w}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{1}{2}, \frac{3}{2};-aw) \)

92

\(\frac{\cos (2\sqrt{a}\sqrt{x})}{\sqrt{x}}\)

\(\frac{1}{\sqrt{w\pi }}{}_{0}F_{2}( ;\frac{1}{2},\frac{1}{2};-aw) \)

93

\(x^{n-\frac{1}{2}}\cos (2\sqrt{a}\sqrt{x})\)

\(\frac{w^{n-\frac{1}{2}}}{( n-\frac{1}{2}) !} {}_{0}F_{2} ( ;\frac{1}{2},n+\frac{1}{2};-aw) \)

94

\(x^{v-1}\cos (2\sqrt{a}\sqrt{x})\)

\(\frac{w^{v-1}}{(v-1)!} {}_{0}F_{2}( ;\frac{1}{2},v;-aw) \)

95

sin(ax)sin(bx)

\(\frac{1}{4} [ \mbox{I}_{0}( 2\sqrt{iw(a-b)}) + \mbox{I}_{0}( 2\sqrt{-iw(a-b)}) ] - \frac{1}{4} [ \mbox{I}_{0}( 2\sqrt{iw(a+b)}) +\mbox{I}_{0}( 2 \sqrt{-iw(a+b)}) ] \)

96

cos(ax)sin(bx)

\(\frac{1}{4} [ i\mbox{I}_{0}( 2\sqrt{iw(a-b)}) -i \mbox{I}_{0}( 2\sqrt{-iw(a-b)}) ] + \frac{1}{4} [ i\mbox{I}_{0}( 2\sqrt{-iw(a+b)}) -\mbox{I}_{0}( 2 \sqrt{iw(a+b)}) ] \)

97

cos(ax)cos(bx)

\(\frac{1}{4} [ \mbox{I}_{0}( 2\sqrt{iw(a+b)}) + \mbox{I}_{0}( 2\sqrt{-iw(a+b)}) ] + \frac{1}{4} [ \mbox{I}_{0}( 2\sqrt{iw(a-b)}) +\mbox{I}_{0}( 2 \sqrt{-iw(a-b)}) ] \)

98

\(\frac{2ax\sin (ax)\cos (ax)-\sin^{2}(ax)}{x^{2}}\)

\(\frac{1}{2\sqrt{a}w^{\frac{5}{2}}} [aw(2aw-1)\operatorname{bei}_{1}( 2\sqrt{2aw}) - aw(2aw+1)\mbox{ber}_{1}( 2 \sqrt{2aw}) - 2(aw)^{\frac{3}{2}}\operatorname{bei}_{0}( 2 \sqrt{2aw}) ] \)

99

\(\frac{ax\cos (ax)-\sin (ax)}{x^{2}}\)

\(\frac{1}{2a^{\frac{3}{2}}w^{\frac{7}{2}}} [ \sqrt{2}(aw)^{2}(aw+1) \operatorname{bei}_{1}( 2\sqrt{aw}) + \sqrt{2}(aw)^{2}(aw-1) \mbox{ber}_{1}( 2\sqrt{aw}) - 2(aw)^{\frac{5}{2}} \mbox{ber}_{0}( 2\sqrt{aw}) ] \)

100

arcsin(x)

\(w{}_{2}F_{3}( \frac{1}{2},\frac{1}{2};1,\frac{3}{2},\frac{3}{2};\frac{w ^{2}}{4}) \)

101

xarcsin(x)

\(\frac{w^{2}}{2} {}_{2}F_{3}( \frac{1}{2},\frac{1}{2};\frac{3}{2}, \frac{3}{2},2;\frac{w^{2}}{4}) \)

102

\(\arctan ( \frac{x}{a})\)

\(\mbox{Si}( \frac{W}{a}) \)

103

\(\cot^{-1}( \frac{x}{a})\)

\(\frac{\pi }{2}-\mbox{Si}( \frac{W}{a}) \)

104

\(x\arctan ( \frac{x}{a})\)

\(a [ \cos ( \frac{x}{a}) -1 ] +w\mbox{Si}( \frac{W}{a}) \)

105

\(x\cot^{-1}( \frac{x}{a})\)

\(a [ 1-\cos ( \frac{x}{a}) ] -w\mbox{Si}( \frac{W}{a}) +\frac{\pi w}{2} \)

106

sinh(ax)

\(\frac{1}{2} [ \mbox{I}_{0}( 2\sqrt{aw}) -\mbox{J}_{0} ( 2\sqrt{aw}) ] \)

107

cosh(ax)

\(\frac{1}{2} [ \mbox{I}_{0}( 2\sqrt{aw}) +\mbox{J}_{0} ( 2\sqrt{aw}) ] \)

108

sinh²(ax)

\(\frac{(aw)^{2}}{2}{}_{1}F_{4}( 1;\frac{3}{2},\frac{3}{2},2,2; \frac{(aw)^{2}}{4}) \)

109

cosh²(ax)

\(1+\frac{(aw)^{2}}{2}{}_{1}F_{4}( 1;\frac{3}{2},\frac{3}{2},2,2; \frac{(aw)^{2}}{4}) \)

110

\(\frac{2\sinh (ax)}{x}\)

\(\sqrt{\frac{a}{w}} [ \mbox{I}_{1}( 2\sqrt{aw}) + \mbox{J}_{1}( 2\sqrt{aw}) ] \)

111

\(\frac{2\cosh (ax)}{x}\)

\(\sqrt{\frac{a}{w}} [ \mbox{I}_{1}( 2\sqrt{aw}) - \mbox{J}_{1}( 2\sqrt{aw}) ] \)

112

\(x^{v-1}\sinh (ax)\)

\(\frac{w^{\frac{v}{2}-1}}{2a^{\frac{v}{2}}} [v\mbox{I}_{v}( 2 \sqrt{aw}) +\sqrt{aw}\mbox{I}_{v+1}( 2\sqrt{aw}) - v\mbox{J}_{v}( 2\sqrt{aw}) +\sqrt{aw}\mbox{J}_{v+1} ( 2\sqrt{aw}) ] \)

113

\(x^{v-1}\cosh (ax)\)

\(\frac{w^{\frac{v-3}{2}}}{2a^{\frac{v+1}{2}}} [ v\sqrt{aw} \mbox{I}_{v}( 2\sqrt{aw}) +aw\mbox{I}_{v+1}( 2 \sqrt{aw}) - v\sqrt{aw}\mbox{J}_{v}( 2\sqrt{aw}) -aw \mbox{J}_{v+1}( 2\sqrt{aw}) ] \)

114

sin(ax)sinh(ax)

\(\frac{(aw)^{2}}{2}{}_{0}F_{7}( ;\frac{3}{4},\frac{3}{4},1, \frac{5}{4},\frac{5}{4}, \frac{3}{2},\frac{3}{2};- \frac{(aw)^{2}}{16\text{,}384}) \)

115

cos(ax)sinh(ax)aw

\({}_{0}F_{7}( ;\frac{1}{2},\frac{1}{2},\frac{3}{4},\frac{3}{4},1, \frac{5}{4},\frac{5}{4};-\frac{(aw)^{2}}{16\text{,}384}) - \frac{(aw)^{3}}{18}{}_{0}F_{7}( ;1,\frac{5}{4},\frac{5}{4}, \frac{3}{2},\frac{3}{2}, \frac{7}{4},\frac{7}{4};- \frac{(aw)^{2}}{16\text{,}384}) \)

116

sin(ax)cosh(ax)

\(aw{}_{0}F_{7}( ;\frac{1}{2},\frac{1}{2},\frac{3}{4},\frac{3}{4},1, \frac{5}{4},\frac{5}{4};-\frac{(aw)^{2}}{16\text{,}384}) + \frac{(aw)^{3}}{18}{}_{0}F_{7}( ;1,\frac{5}{4},\frac{5}{4}, \frac{3}{2},\frac{3}{2}, \frac{7}{4},\frac{7}{4};- \frac{(aw)^{2}}{16\text{,}384}) \)

117

cos(ax)cosh(ax)

\({}_{0}F_{7}( ;\frac{1}{4},\frac{1}{4},\frac{1}{2},\frac{1}{2}, \frac{3}{4},\frac{3}{4},1;-\frac{(aw)^{2}}{16\text{,}384}) \)

118

\(\sinh ( 2\sqrt{a}\sqrt{x})\)

\(\frac{4\sqrt{aw}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{3}{2},\frac{3}{2};aw)\)

119

\(\cosh ( 2\sqrt{a}\sqrt{x})\)

\({}_{0}F_{2}( ;\frac{1}{2},1;aw)\)

120

\(\sqrt{x}\sinh ( 2\sqrt{a}\sqrt{x})\)

\(2\sqrt{a}w{}_{0}F_{2}( ;\frac{3}{2},2;aw) \)

121

\(\sqrt{x}\cosh ( 2\sqrt{a}\sqrt{x})\)

\(\frac{2\sqrt{w}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{1}{2}, \frac{3}{2};aw) \)

122

\(\frac{\sinh ( 2\sqrt{a}\sqrt{x}) }{\sqrt{x}}\)

\(2\sqrt{a}{}_{0}F_{2}( ;1,\frac{3}{2};aw) \)

123

\(\frac{\cosh ( 2\sqrt{a}\sqrt{x}) }{\sqrt{x}}\)

\(\frac{1}{\sqrt{w\pi }}{}_{0}F_{2}( ;\frac{1}{2},\frac{1}{2};aw) \)

124

\(\frac{\sinh^{2}( 2\sqrt{a}\sqrt{x}) }{\sqrt{x}}\)

\(\frac{2a\sqrt{w}}{\sqrt{\pi }}{}_{1}F_{3}( 1;\frac{3}{2}, \frac{3}{2},2;aw) \)

125

\(\frac{\cosh^{2}( 2\sqrt{a}\sqrt{x}) }{\sqrt{x}}\)

\(\frac{1}{\sqrt{\pi w}} [ 1+2aw {}_{1}F_{3}( 1;\frac{3}{2}, \frac{3}{2},2;aw) ] \)

126

\(\frac{\sinh ( 2\sqrt{a}\sqrt{x}) }{x^{\frac{3}{4}}}\)

\(\frac{\sqrt{8a}}{w^{\frac{1}{4}}( -\frac{1}{4}) !} {}_{0}F _{2}( ;\frac{3}{4},\frac{3}{2};2aw) \)

127

\(\frac{\cosh ( 2\sqrt{a}\sqrt{x}) }{x^{\frac{3}{4}}}\)

\(\frac{1}{w^{\frac{3}{4}}( -\frac{1}{4}) !} {}_{0}F_{2}( ; \frac{1}{2},\frac{3}{4};2aw) \)

128

\(x^{v-1}\sinh ( 2\sqrt{a}\sqrt{x})\)

\(\frac{\sqrt{2a}w^{v-\frac{1}{2}}}{( v-\frac{1}{2}) !} {}_{0}F_{2}( ;\frac{3}{2},v+\frac{1}{2};\frac{aw}{2}) \)

129

\(x^{v-1}\cosh ( 2\sqrt{a}\sqrt{x})\)

\(\frac{w^{v-1}}{(v-1)!} {}_{0}F_{2}( ;\frac{1}{2},v;\frac{aw}{2}) \)

130

sinh⁻¹(x)

\(w{}_{2}F_{3}( \frac{1}{2},\frac{1}{2};1,\frac{3}{2}, \frac{3}{2};-\frac{w ^{2}}{4}) \)

131

cosh⁻¹(x)

\(-iw{}_{2}F_{3}( \frac{1}{2},\frac{1}{2};1,\frac{3}{2}, \frac{3}{2};\frac{w^{2}}{4}) +\frac{i\pi }{2} \)

132

\(\cosh^{-1}( 1+\frac{x}{a})\)

\(\sqrt{\frac{8w}{a\pi }}{}_{2}F_{2}( \frac{1}{2},\frac{1}{2}; \frac{3}{2}, \frac{3}{2};-\frac{w}{2a}) \)

133

xsinh⁻¹(x)

\(\frac{w^{2}}{2}{}_{2}F_{3}( \frac{1}{2},\frac{1}{2};\frac{3}{2}, \frac{3}{2},2;-\frac{w^{2}}{4}) \)

134

sinh((2n + 1)sinh⁻¹(x))

\((2n+1)w{}_{2}F_{3}( -n,n+1;1,\frac{3}{2}, \frac{3}{2};-\frac{w ^{2}}{4}) \)

135

cosh(2nsinh⁻¹(x))

\({}_{2}F_{3}( n,-n;\frac{1}{2}, \frac{1}{2},1;-\frac{w^{2}}{4}) \)

136

sinh(vsinh⁻¹(x))

\(vw{}_{2}F_{3}( \frac{1+v}{2},\frac{1-v}{2};1,\frac{3}{2}, \frac{3}{2};-\frac{w^{2}}{4}) \)

137

cosh(vsinh⁻¹(x))

\({}_{2}F_{3}( \frac{v}{2},-\frac{v}{2};\frac{1}{2}, \frac{1}{2},1;-\frac{w ^{2}}{4}) \)

138

\(\sinh ( v\cosh^{-1}( 1+\frac{x}{a}) )\)

\(v\sqrt{\frac{8w}{a\pi }}{}_{2}F_{2}( \frac{1}{2}+v,\frac{1}{2}-v; \frac{3}{2}, \frac{3}{2};-\frac{w}{2a}) \)

139

\(\frac{\exp (n\sinh^{-1}(x))}{\sqrt{1+x^{2}}}\)

\(nw {}_{2}F_{3}( 1+\frac{n}{2},1-\frac{n}{2};1,\frac{3}{2}, \frac{3}{2}; -\frac{w^{2}}{4}) + {}_{2}F_{3}( \frac{1-n}{2}, \frac{1+n}{2};\frac{1}{2},\frac{1}{2},1; -\frac{w^{2}}{4}) \)

140

\(\frac{\exp (-n\sinh^{-1}(x))}{\sqrt{1+x^{2}}}\)

\(-nw {}_{2}F_{3}( 1-\frac{n}{2},1+\frac{n}{2};1,\frac{3}{2}, \frac{3}{2}; -\frac{w^{2}}{4}) + {}_{2}F_{3}( \frac{1-n}{2}, \frac{1+n}{2};\frac{1}{2},\frac{1}{2},1; -\frac{w^{2}}{4}) \)

141

\(\frac{\sinh (v\sinh^{-1}(x))}{\sqrt{x^{2}+1}}\)

\(vw {}_{2}F_{3}( 1-\frac{v}{2},1+\frac{v}{2};1,\frac{3}{2}, \frac{3}{2}; -\frac{w^{2}}{4}) \)

142

\(\frac{\cosh (n\sinh^{-1}(x))}{\sqrt{x^{2}+1}}\)

\({}_{2}F_{3}( \frac{1-n}{2},\frac{1+n}{2};\frac{1}{2},\frac{1}{2},1; -\frac{w^{2}}{4}) \)

143

\(\frac{\cosh (n\cosh^{-1}(x))}{\sqrt{x^{2}-1}}\)

\(-\frac{1}{2e^{\frac{in\pi }{2}}} [ i( 1+e^{in\pi }) {}_{2}F_{3}( \frac{1-n}{2},\frac{1+n}{2};\frac{1}{2},\frac{1}{2},1;\frac{w ^{2}}{4}) + nw( 1+e^{in\pi }) {}_{2}F_{3}( 1- \frac{n}{2},1+\frac{n}{2};1,\frac{3}{2},\frac{3}{2};\frac{w^{2}}{4}) ] \)

144

\(\frac{\exp (2v\sinh^{-1}( \frac{x}{2a}) )}{\sqrt{x^{3}+4a ^{2}x}}\)

\(\frac{\operatorname{csgn}(a)}{2a^{2}\sqrt{\pi w}} [ a{}_{2}F_{3}( \frac{1}{2}+v, \frac{1}{2}-v;\frac{1}{4},\frac{1}{2},\frac{3}{4};-\frac{w^{2}}{16a ^{2}}) + 2vw{}_{2}F_{3}( 1-v,1+v;\frac{3}{4},\frac{5}{4}, \frac{3}{2};-\frac{w^{2}}{16a^{2}}) ] \)

145

\(\frac{\exp (-2v\sinh^{-1}( \frac{x}{2a}) )}{\sqrt{x^{3}+4a ^{2}x}}\)

\(\frac{\operatorname{csgn}(a)}{2a^{2}\sqrt{\pi w}} [ a{}_{2}F_{3}( \frac{1}{2}+v, \frac{1}{2}-v;\frac{1}{4},\frac{1}{2},\frac{3}{4};-\frac{w^{2}}{16a ^{2}}) - 2vw{}_{2}F_{3}( 1-v,1+v;\frac{3}{4},\frac{5}{4}, \frac{3}{2};-\frac{w^{2}}{16a^{2}}) ] \)

146

\(\frac{1}{\sqrt{x^{3}+4a^{2}x}} (\cos ( ( v+\frac{1}{4})\pi ) \exp ( -2v\sinh^{-1}( \frac{x}{2a}) )+\sin ( ( v+\frac{1}{4}) \pi ) \exp ( 2v\sinh^{-1}( \frac{x}{2a}) ) )\)

\(\frac{\operatorname{csgn}(a)}{a^{2}\sqrt{2\pi w}} [a\cos (\pi v){}_{2}F_{3}( \frac{1}{2}+v,\frac{1}{2}-v;\frac{1}{4},\frac{1}{2}, \frac{3}{4};-\frac{w^{2}}{16a^{2}}) + 2vw\sin (\pi v){}_{2}F_{3}( 1-v,1+v;\frac{3}{4},\frac{5}{4},\frac{3}{2};-\frac{w^{2}}{16a ^{2}}) ] \)

147

\(\frac{1}{\sqrt{x^{3}+4a^{2}x}} (\sin ( ( v+\frac{1}{4})\pi ) \exp ( -2v\sinh^{-1}( \frac{x}{2a}) )-\cos ( ( v+\frac{1}{4}) \pi ) \exp ( 2v\sinh^{-1}( \frac{x}{2a}) ) ) \)

\(\frac{\operatorname{csgn}(a)}{a^{2}\sqrt{2\pi w}} [ a\sin (\pi v){} _{2}F_{3}( \frac{1}{2}+v,\frac{1}{2}-v;\frac{1}{4},\frac{1}{2}, \frac{3}{4};-\frac{w^{2}}{16a^{2}}) - 2vw\cos (\pi v){}_{2}F _{3}( 1-v,1+v;\frac{3}{4},\frac{5}{4},\frac{3}{2};-\frac{w^{2}}{16a ^{2}}) ] \)

148

\(\frac{\sinh ( 2v\sinh^{-1}( \frac{x}{2a}) ) }{\sqrt{x^{3}+4a^{2}x}}\)

\(\frac{v\sqrt{w}\operatorname{csgn}(a)}{a^{2}\sqrt{\pi }}{}_{2}F_{3}( 1-v,1+v; \frac{3}{4},\frac{5}{4},\frac{3}{2};-\frac{w^{2}}{16a^{2}}) \)

149

\(\frac{\cosh ( 2v\sinh^{-1}( \frac{x}{2a}) ) }{\sqrt{x^{3}+4a^{2}x}}\)

\(\frac{\operatorname{csgn}(a)}{2a\sqrt{\pi w}}{}_{2}F_{3}( \frac{1}{2}+v,\frac{1}{2}-v;\frac{1}{4},\frac{1}{2},\frac{3}{4};-\frac{w^{2}}{16a ^{2}})\)

150

\(\frac{\sinh^{-1}(x)}{x}\)

\(\mbox{J}_{0}(w)-\frac{\pi }{2} [ \mbox{J}_{0}(w)\textbf{H} _{1}(w) -\mbox{J}_{1}(w)\textbf{H}_{0}(w) ]\)