Many elementary and non-elementary transcendental functions may be expressed as special cases of the hypergeometric functions $$F(a,\,b,\,c;\,x) \;=\; 1+\frac{ab}{1!c}x+\frac{a(a+1)b(b+1)}{2!c(c+1)}x^2 +\frac{a(a+1)(a+2)b(b+1)(b+2)}{3!c(c+1)(c+2)}x^3+\ldots,$$ which are solutions of the hypergeometric equation $$x(x-1)\frac{d^2y}{dx^2}+(c-(a+b+1))\frac{dy}{dx}-aby \;=\;0.$$ For example:

• $(1\!+\!x)^n \;=\; F(-n,\,1,\,1;\,-x)$
• $\ln(1\!+\!x) \;=\; xF(1,\,1,\,2;\,-x)$
• $\ln\frac{1+x}{1-x} \;=\; 2xF(\frac{1}{2},\,1,\,\frac{3}{2};\,x^2)$
• $\arcsin{x} \;=\; xF(\frac{1}{2},\,\frac{1}{2},\,\frac{3}{2};\,x^2)$
• $\arctan{x} \;=\; xF(\frac{1}{2},\,1,\,\frac{3}{2};\,-x^2)$
• $\sin(m\arcsin{x}) \;=\; mxF(\frac{1+m}{2},\,\frac{1-m}{2},\,\frac{3}{2};\,x^2)$
• $\cos(m\arcsin{x}) \;=\; F(\frac{m}{2},\,-\frac{m}{2},\,\frac{1}{2};\,x^2)$
• $T_n(x) \;=\; F(n,\,-n,\,\frac{1}{2};\,\frac{1-x}{2})$ (Chebyshev polynomials)
• $P_n(x) \;=\; F(-n,\,n+1,\,1;\,\frac{1-x}{2})$ (Legendre polynomials)
• $\displaystyle\int_0^{\frac{\pi}{2}}\!\frac{d\varphi}{\sqrt{1\!-\!x^2\sin^2\varphi}}$ $\;=\; \frac{\pi}{2}F(\frac{1}{2},\,\frac{1}{2},\,1;\,x^2)$ (complete elliptic integral of 1st kind)
• $\displaystyle\int_0^{\frac{\pi}{2}}\!\sqrt{1\!-\!x^2\sin^2\varphi}\;d\varphi$ $\;=\; \frac{\pi}{2}F(-\frac{1}{2},\,\frac{1}{2},\,1;\,x^2)$ (complete elliptic integral of 2nd kind)