Fungsi hipergeometris

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Fungsi hipergeometris
Fungsi hipergeometris biasa	2F1(a,b;c;z)
Deret hipergeometris	:${\displaystyle {}_{2}F_{1}(a,b;c;z)=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}}.}$
Rumus Antiturunan	: ${\displaystyle {\frac {d}{dz}}\ {}_{2}F_{1}(a,b;c;z)={\frac {ab}{c}}\ {}_{2}F_{1}(a+1,b+1;c+1;z)}$ dan lebih umum ${\displaystyle {\frac {d^{n}}{dz^{n}}}\ {}_{2}F_{1}(a,b;c;z)={\frac {(a)_{n}(b)_{n}}{(c)_{n}}}\ {}_{2}F_{1}(a+n,b+n;c+n;z)}$ In the special case that ${\displaystyle c=a+1}$, we have ${\displaystyle {\frac {d}{dz}}\ {}_{2}F_{1}(a,b;a+1;z)={\frac {d}{dz}}\ {}_{2}F_{1}(b,a;a+1;z)={\frac {a((1-z)^{-b}-{}_{2}F_{1}(a,b;1+a;z))}{z}}}$
Persamaan turunan Fungsi hipergeometris	:${\displaystyle z(1-z){\frac {d^{2}w}{dz^{2}}}+\left[c-(a+b+1)z\right]{\frac {dw}{dz}}-ab\,w=0.}$
Pecahan berlanjut Gauus	:${\displaystyle {\frac {{}_{2}F_{1}(a+1,b;c+1;z)}{{}_{2}F_{1}(a,b;c;z)}}={\cfrac {1}{1+{\cfrac {{\frac {(a-c)b}{c(c+1)}}z}{1+{\cfrac {{\frac {(b-c-1)(a+1)}{(c+1)(c+2)}}z}{1+{\cfrac {{\frac {(a-c-1)(b+1)}{(c+2)(c+3)}}z}{1+{\cfrac {{\frac {(b-c-2)(a+2)}{(c+3)(c+4)}}z}{1+{}\ddots }}}}}}}}}}}$

Dalam matematika, Fungsi hipergeometris biasa atau Gaussia ₂F₁(a,b;c;z) adalah sebuah fungsi istimewa yang diwakili oleh rangkaian hipergeometris, yang meliputi sebagian besar fungsi istimewa lainnya sebagai kasus spesifik atau pembatasan. Fungsi tersebut adalah solusi dari persamaan diferensial biasa (ODE) linear urutan kedua. Setiap ODE liberal urutan kedua dengan tiga titik tinggal reguler dapat bertransformasi menjadi persamaan tersebut.

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