Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 31/Special Cases
Hypergeometric Functions
$31.3$: Power of One plus x in terms of Gaussian Hypergeometric Function
- $\map F {-p, 1; 1; -x} = \paren {1 + x}^p$
$31.4$: Logarithm of One plus x in terms of Gaussian Hypergeometric Function
- $\map \ln {1 + x} = x \map F {1, 1; 2; -x}$
$31.5$: Exponential Function as Limit of Gaussian Hypergeometric Function
Exponential Function as Limit of Gaussian Hypergeometric Function
$31.6$: Cosine Function in terms of Gaussian Hypergeometric Function Cosine Function in terms of Gaussian Hypergeometric Function
$31.7$: Secant Function in terms of Gaussian Hypergeometric Function Secant Function in terms of Gaussian Hypergeometric Function
$31.8$: Arcsine Function in terms of Gaussian Hypergeometric Function
- $\arcsin x = x \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; x^2}$
$31.9$: Arctangent Function in terms of Gaussian Hypergeometric Function
- $\arctan x = x \map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2}$
$31.10$: Reciprocal of One minus x in terms of Gaussian Hypergeometric Function
- $\dfrac 1 {1 - x} = \map F {1, p; p; x}$
$31.11$: Legendre Polynomial in terms of Gaussian Hypergeometric Function
Legendre Polynomial in terms of Gaussian Hypergeometric Function
$31.12$: Chebyshev Polynomial of First Kind in terms of Gaussian Hypergeometric Function Chebyshev Polynomial of First Kind in terms of Gaussian Hypergeometric Function