Theory and Modern Applications
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 |