複素数

冪乗の対数

by nomura · 2021年9月12日

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冪乗の対数

(1)

Log e^{z} = ℜ z + i \mod (ℑ z, - 2 π, π)

(2)

Log {| α |}^{β} = ℜ (β) \ln | α | + i \mod (ℑ (β) \ln | α |, - 2 π, π)

(3)

Log α^{β} = ℜ (β) \ln | α | - ℑ (β) Arg (α) + \mod (ℜ (β) Arg (α) + ℑ (β) \ln | α |, - 2 π, π)

(1)

\begin{aligned} Log e^{z} & = Log e^{ℜ z + i ℑ z} \\ = \ln e^{ℜ z} + Log e^{i ℑ z} \\ = ℜ z + \ln | e^{i ℑ z} | + i Arg e^{i ℑ z} \\ = ℜ z + i \mod (ℑ z, - 2 π, π) \end{aligned}

(2)

\begin{aligned} Log {| α |}^{β} & = Log {| α |}^{ℜ (β) + i ℑ (β)} \\ = \ln | {| α |}^{ℜ (β) + i ℑ (β)} | + i Arg ({| α |}^{ℜ (β) + i ℑ (β)}) \\ = \ln | {| α |}^{ℜ (β)} {| α |}^{i ℑ (β)} | + i Arg ({| α |}^{ℜ (β)} {| α |}^{i ℑ (β)}) \\ = \ln {| α |}^{ℜ (β)} + i Arg ({| α |}^{i ℑ (β)}) \\ = \ln {| α |}^{ℜ (β)} + i Arg (e^{i ℑ (β) \ln | α |}) \\ = ℜ (β) \ln | α | + i \mod (ℑ (β) \ln | α |, - 2 π, π) \end{aligned}

(2)-2

\begin{aligned} Log {| α |}^{β} & = Log {| α |}^{| β | e^{i Arg (β)}} \\ = \ln | {| α |}^{| β | e^{i Arg (β)}} | + i Arg ({| α |}^{| β | e^{i Arg (β)}}) \\ = \ln \sqrt{{| α |}^{| β | e^{i Arg (β)}} {| α |}^{| β | e^{- i Arg (β)}}} + i Arg ({| α |}^{| β | e^{i Arg (β)}}) \\ = \ln \sqrt{{| α |}^{| β | (e^{i Arg (β)} + e^{- i Arg (β)})}} + i Arg (e^{| β | e^{i Arg (β)} Log | α |}) \\ = \ln \sqrt{{| α |}^{2 | β | \cos (Arg β)}} + i Arg (e^{| β | Log | α | (\cos (Arg β) + i \sin (Arg β))}) \\ = \ln {| α |}^{| β | \cos (Arg β)} + i Arg (e^{| β | Log | α | \cos (Arg β) + i | β | Log | α | \sin (Arg β)}) \\ = | β | \cos (Arg β) \ln | α | + i Arg (e^{| β | Log | α | \cos (Arg β)} e^{i | β | Log | α | \sin (Arg β)}) \\ = | β | \cos (Arg β) \ln | α | + i \mod (| β | \sin (Arg β) Log | α |, - 2 π, π) \\ = ℜ (β) \ln | α | + i \mod (ℑ (β) Log | α |, - 2 π, π) \end{aligned}

(3)

\begin{aligned} Log α^{β} & = Log e^{β Log α} \\ = Log e^{(ℜ (β) + i ℑ (β)) (\ln | α | + i Arg (α))} \\ = Log (e^{ℜ (β) \ln | α | - ℑ (β) Arg (α)} e^{i (ℜ (β) Arg (α) + ℑ (β) \ln | α |)}) \\ = \ln e^{ℜ (β) \ln | α | - ℑ (β) Arg (α)} + i Arg e^{i (ℜ (β) Arg (α) + ℑ (β) \ln | α |)} \\ = ℜ (β) \ln | α | - ℑ (β) Arg (α) + \mod (ℜ (β) Arg (α) + ℑ (β) \ln | α |, - 2 π, π) \end{aligned}

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指数関数の実部と虚部

| α^{β} | = {| α |}^{ℜ (β)} e^{- ℑ (β) Arg (α)}

偏角・対数と絶対値

Log (| α | β) = \ln | α | + Log β

複素数と複素共役の和・差

z \pm \overset{―}{z} = 2 H (\pm 1) ℜ z + 2 i H (\mp 1) ℑ z

絶対値の冪乗

{(| α | β)}^{γ} = {| α |}^{γ} β^{γ}