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Next: Trigonometric identities Up: Useful mathematics Previous: Calculus

Series expansions

Notation: $k!= k\,(k-1)\,(k-2)..2.1$, $f^{(n)}(x)=d^{\,n}f(x)/dx^{\,n}$.

$\displaystyle f(x)$ $\displaystyle = f(a) + \frac{(x-a)}{1!}\,f^{(1)}(a) + \frac{(x-a)^2}{2!}\,f^{(2)}(a)+\cdots \frac{(x-a)^n}{n!}\,f^{(n)}(a)+\cdot$ (A.15)
$\displaystyle (1+x)^\alpha$ $\displaystyle = 1 + \alpha\,x + \frac{\alpha\,(\alpha-1)}{2!}\,x^{\,2} + \frac{\alpha\,(\alpha-1)\,(\alpha-2)}{3!} \,x^{\,3}+\cdots$ (A.16)
$\displaystyle {\rm e}^x$ $\displaystyle = 1 + x + \frac{x^{\,2}}{2!}+ \frac{x^{\,3}}{3!} + \cdots$ (A.17)
$\displaystyle \ln(1+x)$ $\displaystyle = x - \frac{x^{\,2}}{2}+\frac{x^{\,3}}{3}-\frac{x^{\,4}}{4}+\cdots$ (A.18)
$\displaystyle \sin x$ $\displaystyle = x - \frac{x^{\,3}}{3!} +\frac{x^{\,5}}{5!} - \frac{x^{\,7}}{7!} + \cdots$ (A.19)
$\displaystyle \cos x$ $\displaystyle = 1-\frac{x^{\,2}}{2!} + \frac{x^{\,4}}{4!}-\frac{x^{\,6}}{6!} + \cdots$ (A.20)
$\displaystyle \tan x$ $\displaystyle = x + \frac{x^{\,3}}{3} + \frac{2\,x^{\,5}}{15} + \frac{17\,x^{\,7}}{315}+\cdots$ (A.21)
$\displaystyle \sinh x$ $\displaystyle = x + \frac{x^{\,3}}{3!} +\frac{x^{\,5}}{5!} + \frac{x^{\,7}}{7!} + \cdots$ (A.22)
$\displaystyle \cosh x$ $\displaystyle = 1+\frac{x^{\,2}}{2!} + \frac{x^{\,4}}{4!}+\frac{x^{\,6}}{6!} + \cdots$ (A.23)
$\displaystyle \tanh x$ $\displaystyle = x - \frac{x^{\,3}}{3} + \frac{2\,x^{\,5}}{15} - \frac{17\,x^{\,7}}{315}+\cdots$ (A.24)