簡約 ln(((i/2-1/2)!)/(sqrt(pi)(i/2)!))

解

簡約

ln (\frac{( \frac{i}{2} - \frac{1}{2} ) !}{π ( \frac{i}{2} ) !})

ln (\frac{i - 1}{2}!) - \frac{1}{2} ln (π) - ln (\frac{i}{2}!)

解答ステップ

ln (\frac{( \frac{i}{2} - \frac{1}{2} ) !}{π ( \frac{i}{2} ) !})

対数の規則を適用する:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{( \frac{i}{2} - \frac{1}{2} ) !}{π ( \frac{i}{2} ) !}) = ln ((\frac{i}{2} - \frac{1}{2})!) - ln (π (\frac{i}{2})!)

= ln ((- \frac{1}{2} + \frac{i}{2})!) - ln (π \frac{i}{2}!)

対数の規則を適用する:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln (π \frac{i}{2}!) = ln (π) + ln (\frac{i}{2}!)

= ln ((- \frac{1}{2} + \frac{i}{2})!) - (ln (π) + ln (\frac{i}{2}!))

ln (π) = \frac{1}{2} ln (π)

= ln ((- \frac{1}{2} + \frac{i}{2})!) - (\frac{1}{2} ln (π) + ln (\frac{i}{2}!))

- (\frac{1}{2} ln (π) + ln (\frac{i}{2}!)) : - \frac{1}{2} ln (π) - ln (\frac{i}{2}!)

= ln ((\frac{i}{2} - \frac{1}{2})!) - \frac{1}{2} ln (π) - ln (\frac{i}{2}!)

分数を組み合わせる

\frac{i}{2} - \frac{1}{2} : \frac{i - 1}{2}

= ln (\frac{- 1 + i}{2}!) - \frac{1}{2} ln (π) - ln (\frac{i}{2}!)

= ln (\frac{i - 1}{2}!) - \frac{1}{2} ln (π) - ln (\frac{i}{2}!)