special cases of hypergeometric function

Many elementary (http://planetmath.org/ElementaryFunction) and non-elementary transcendental functions may be expressed as special cases of the hypergeometric functions

$$F(a,b,c;x)=\mathrm{\hspace{0.33em}1}+\frac{ab}{1!c}x+\frac{a(a+1)b(b+1)}{2!c(c+1)}{x}^2+\frac{a(a+1)(a+2)b(b+1)(b+2)}{3!c(c+1)(c+2)}{x}^3+\mathrm{\dots },$$

which are solutions of the hypergeometric equation

$$x(x-1)\frac{d^{2}y}{d{x}^2}+(c-(a+b+1))\frac{dy}{dx}-aby=\mathrm{\hspace{0.33em}0}.$$

For example:

• ${(1+x)}^n=F(-n,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}1};-x)$

• $\ln (1+x)=xF(1,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2};-x)$

• $\ln \frac{1+x}{1-x}=\mathrm{\hspace{0.33em}2}xF(\frac{1}{2},\mathrm{\hspace{0.17em}1},\frac{3}{2};x^2)$

• $\arcsin x=xF(\frac{1}{2},\frac{1}{2},\frac{3}{2};x^2)$

• $\arctan x=xF(\frac{1}{2},\mathrm{\hspace{0.17em}1},\frac{3}{2};-x^2)$

• $\sin (m\arcsin x)=mxF(\frac{1+m}{2},\frac{1-m}{2},\frac{3}{2};x^2)$

• $\cos (m\arcsin x)=F(\frac{m}{2},-\frac{m}{2},\frac{1}{2};x^2)$

• $T_n(x)=F(n,-n,\frac{1}{2};\frac{1-x}{2})$  (Chebyshev polynomials)

• $P_n(x)=F(-n,n+1,\mathrm{\hspace{0.17em}1};\frac{1-x}{2})$  (Legendre polynomials)

• ${\displaystyle {\int }_0^{\frac{\pi }{2}}}{\displaystyle \frac{d\phi }{\sqrt{1-x^2{\sin }^2\phi }}}$$=\frac{\pi }{2}F(\frac{1}{2},\frac{1}{2},\mathrm{\hspace{0.17em}1};x^2)$  (complete elliptic integral of 1st kind)

• ${\displaystyle {\int }_0^{\frac{\pi }{2}}}\sqrt{1-x^2{\sin }^2\phi }𝑑\phi$$=\frac{\pi }{2}F(-\frac{1}{2},\frac{1}{2},\mathrm{\hspace{0.17em}1};x^2)$  (complete elliptic integral of 2nd kind)