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Table of contents: Integer exponents - Elementary functions - Complex parts

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Symbol: Pow ab{a}^{b} Power

Integer exponents

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00=1{0}^{0} = 1
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z0=1{z}^{0} = 1
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z1=z{z}^{1} = z
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zn+1=znz{z}^{n + 1} = {z}^{n} z
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z1=1z{z}^{-1} = \frac{1}{z}

Elementary functions

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ab=exp ⁣(blog ⁣(a)){a}^{b} = \exp\!\left(b \log\!\left(a\right)\right)
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z1/2=z{z}^{1 / 2} = \sqrt{z}
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z1/2=1z{z}^{-1 / 2} = \frac{1}{\sqrt{z}}

Complex parts

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(a+bi)c+di=Mcedθ(cos ⁣(cθ+dlog ⁣(M))+isin ⁣(cθ+dlog ⁣(M)))   where M=a+bi,θ=arg ⁣(a+bi){\left(a + b i\right)}^{c + d i} = {M}^{c} {e}^{-d \theta} \left(\cos\!\left(c \theta + d \log\!\left(M\right)\right) + i \sin\!\left(c \theta + d \log\!\left(M\right)\right)\right)\; \text{ where } M = \left|a + b i\right|,\,\theta = \arg\!\left(a + b i\right)
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(a+bi)c+di=Mcedθ   where M=a+bi,θ=arg ⁣(a+bi)\left|{\left(a + b i\right)}^{c + d i}\right| = {M}^{c} {e}^{-d \theta}\; \text{ where } M = \left|a + b i\right|,\,\theta = \arg\!\left(a + b i\right)
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Re ⁣((a+bi)c+di)=Mcedθcos ⁣(cθ+dlog ⁣(M))   where M=a+bi,θ=arg ⁣(a+bi)\operatorname{Re}\!\left({\left(a + b i\right)}^{c + d i}\right) = {M}^{c} {e}^{-d \theta} \cos\!\left(c \theta + d \log\!\left(M\right)\right)\; \text{ where } M = \left|a + b i\right|,\,\theta = \arg\!\left(a + b i\right)
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Im ⁣((a+bi)c+di)=Mcedθsin ⁣(cθ+dlog ⁣(M))   where M=a+bi,θ=arg ⁣(a+bi)\operatorname{Im}\!\left({\left(a + b i\right)}^{c + d i}\right) = {M}^{c} {e}^{-d \theta} \sin\!\left(c \theta + d \log\!\left(M\right)\right)\; \text{ where } M = \left|a + b i\right|,\,\theta = \arg\!\left(a + b i\right)

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2019-05-23 08:00:13.607731 UTC