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Theory and Modern Applications

Table 2 Special functions definition

From: Computing new solutions of algebro-geometric equation using the discrete inverse Sumudu transform

S. No

Function

Definition

1

First kind Bessel function

\(\mbox{J}_{n}(x)=\sum_{k=0}^{\infty }\frac{(-1)^{k}( \frac{x}{2})^{2k+n}}{k!(n+k)!}\)

2

Modified first kind Bessel function

\(\mbox{I}_{n}(x)=\sum_{k=0}^{\infty }\frac{( \frac{x}{2}) ^{2k+n}}{k!(n+k)!} \)

3

Kelvin real function

\(\mbox{ber}_{n}(x)=\operatorname{Re} J_{n}(i^{\frac{3}{2}}x) \)

4

Kelvin imaginary function

\(\operatorname{bei}_{n}(x)=\operatorname{Im} J_{n}(i^{\frac{3}{2}}x) \)

5

Error function

\(\mbox{erf}(x)=\frac{2}{\sqrt{\pi }}\int_{0}^{x}e^{-z^{2}}\,dz \)

6

Complementary error function

\(\mbox{erfc}(x)=\frac{2}{\sqrt{\pi }}\int_{x}^{\infty }e^{-z^{2}}\,dz \)

7

Struve function

\(\textbf{H}_{v}(x)=( \frac{x}{2}) ^{v+1} \sum_{k=0} ^{\infty }\frac{(-1)^{k}( \frac{x}{2}) ^{2k}}{\Gamma ( k+ \frac{3}{2}) \Gamma ( k+v+\frac{3}{2}) } \)

8

Modified Struve function

\(\textbf{L}_{v}(x)=( \frac{x}{2}) ^{v+1} \sum_{k=0} ^{\infty }\frac{( \frac{x}{2}) ^{2k}}{\Gamma ( k+ \frac{3}{2}) \Gamma ( k+v+\frac{3}{2}) } \)

9

Generalized hypergeometric function

\({}_{p}F_{q}( (a_{p});(b_{q});x) =\sum_{k=0}^{\infty }\frac{(a_{1})_{k}\cdot(a_{2})_{k}\cdots (a_{p})_{k}x^{k}}{(b_{1})_{k}\cdot(b_{2})_{k} \cdots (b_{q})_{k}k!} \)

10

Lommel S1 function

\(\textbf{S}^{(1)}_{\mu ,v}(x)=\frac{x^{\mu +1}{}_{1}F_{2} ( 1;\frac{ \mu -v+3}{2},\frac{\mu +v+3}{2};-\frac{x^{2}}{4}) }{(\mu +1)^{2}-v ^{2}} \)

11

Whittaker M function

\(\mbox{M}_{\kappa ,\mu }(x)=e^{-\frac{x}{2}}x^{\mu +\frac{1}{2}} \mbox{M} ( \mu -\kappa +\frac{1}{2},1+2\mu ;x) \)

12

Kummer function

\(\mbox{M}(a,b,c)=\sum_{n=0}^{\infty }\frac{a^{(n)}x^{n}}{b^{(n)}n!}= {}_{1}F_{1}(a;b;x) \)

13

Sign function

\(\operatorname{csgn}(x)= \begin{cases} 1 ; x<\mathscr{R}(x), \\ -1 ; x>\mathscr{R}(x) \end{cases} \)

14

Sine integral

\(\mbox{Si}(x)=\int_{0}^{x}\frac{\sin (z)\,dz}{z} \)

15

Cosine integral

\(\mbox{Ci}(x)=-\int_{x}^{\infty }\frac{\cos (z)\,dz}{z} \)

16

Hyperbolic cosine integral

\(\mbox{Chi}(x)=\gamma +\ln (x)+\int_{x}^{\infty }\frac{(\cos (z)-1)\,dz}{z} \)

17

Exponential integral

\(\mbox{Ei}(x)=\int_{x}^{\infty }\frac{e^{-z}\,dz}{z} \)

18

Laguerre polynomials

\(\mbox{L}_{n}(x)=\frac{e^{x}}{n!}\frac{d^{n}}{dx^{n}}( x^{n}e^{-x}) \)

19

Euler’s constant

γ = 0.5772156