توان (ریاضی) - ویکی‌پدیا، دانشنامهٔ آزاد

${\displaystyle b^{n}=\underbrace {b\times \cdots \times b} _{n}}$

a¹ = a

a⁰ = 1

a⁻¹ = 1/a

a⁻ⁿ = (aⁿ)⁻¹ = 1/aⁿ

${\displaystyle a^{m+n}=a^{m}\cdot a^{n}}$

${\displaystyle a^{m-n}={\begin{matrix}{\frac {a^{m}}{a^{n}}}\end{matrix}}}$

${\displaystyle (a^{m})^{n}=a^{mn}\!\,}$

${\displaystyle i^{4n+1}=i}$

${\displaystyle i^{4n+2}=-1}$

${\displaystyle i^{4n+3}=-i}$

${\displaystyle i^{4n}=1}$

${\displaystyle \ e=\lim _{n\rightarrow +\infty }\left(1+{\frac {1}{n}}\right)^{n}=\lim _{n\rightarrow -\infty }\left(1+{\frac {1}{n}}\right)^{n}}$.

${\displaystyle \ e\approx 2.71828}$.

${\displaystyle e^{x}=\left(\lim _{m\rightarrow \pm \infty }\left(1+{\frac {1}{m}}\right)^{m}\right)^{x}=\lim _{m\rightarrow \pm \infty }\left(\left(1+{\frac {1}{m}}\right)^{m}\right)^{x}=\lim _{m\rightarrow \pm \infty }\left(1+{\frac {1}{m}}\right)^{mx}=\lim _{mx\rightarrow \pm \infty }\left(1+{\frac {x}{mx}}\right)^{mx}=\lim _{n\rightarrow \pm \infty }\left(1+{\frac {x}{n}}\right)^{n}}$

${\displaystyle \ x^{n}=a}$

${\displaystyle x=a^{\frac {1}{n}}}$

z⁰ = 1 zⁿ⁺¹ = z·zⁿ z⁻ⁿ = 1/zⁿ (برای z ≠ 0)

${\displaystyle e^{z}=\lim _{n\rightarrow \infty }\left(1+{\frac {z}{n}}\right)^{n}}$

a^z = e^bz

a = e^b

${\displaystyle \ e^{ix}=\cos(x)+i\sin(x)}$ ${\displaystyle \ e^{-ix}=\cos(x)-i\sin(x)}$

${\displaystyle \ \cos(x)=(e^{ix}+e^{-ix})/{2}}$ ${\displaystyle \ \sin(x)=(e^{ix}-e^{-ix})/{2i}}$

${\displaystyle a+ib=re^{i\varphi }=r\left[\cos \varphi +i\sin \varphi \right]}$

${\displaystyle (a+ib)^{x}=\left(re^{i\varphi }\right)^{x}=r^{x}e^{i\varphi x}.}$

${\displaystyle r^{id}=\left[(r)^{d}\right]^{i}=\left[\left(e^{\ln r}\right)^{d}\right]^{i}=e^{id\ln r}=\cos(d\ln r)+i\sin(d\ln r).}$

${\displaystyle (a+ib)^{c+id}=\left(re^{i\varphi }\right)^{c+id}=\left[r^{c}e^{-\varphi d}\right]e^{i(\varphi c+d\ln r)}}$

${\displaystyle i^{i}=(e^{i\pi /2})^{i}=e^{-\pi /2}\approx 0.20788\ldots }$

${\displaystyle i^{i}=(e^{i\pi /2+2\pi i\cdot n})^{i}=e^{-\pi /2-2\pi \cdot n}}$

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