式展開: ガンマ凾数の相反公式

ガンマ凾数の相反公式

ガンマ凾数の相反公式

\begin{array}{rcl} Γ (z) Γ (1 - z) & = & Γ (z) Γ (- z + 1) \dots Re (z) > 0 か つ Re (1 - z) > 0 な の で 0 < Re (z) < 1 \\ = & Γ (z) (- z Γ (- z)) \dots Γ (z + 1) = z Γ (z) \\ = & - z Γ (z) Γ (- z) \\ = & - z (lim_{n \to \infty} \frac{n^{z} n!}{\prod_{k = 0}^{n} (z + k)}) (lim_{n \to \infty} \frac{n^{- z} n!}{\prod_{k = 0}^{n} (- z + k)}) \dots Γ (z) = lim_{n \to \infty} \frac{n^{z} n!}{\prod_{k = 0}^{n} (z + k)} \\ = & - z (lim_{n \to \infty} \frac{n^{z} n! \cdot n^{- z} n!}{\prod_{k = 0}^{n} (z + k) (- z + k)}) \\ = & - z (lim_{n \to \infty} \frac{n! n!}{\prod_{k = 0}^{n} (k^{2} - z^{2})}) \dots A^{B} A^{- B} = A^{B - B} = A^{0} = 1 \\ = & - z (lim_{n \to \infty} \frac{(\prod_{k = 1}^{n} k) (\prod_{k = 1}^{n} k)}{\prod_{k = 0}^{n} (k^{2} - z^{2})}) \dots n! = \prod_{k = 1}^{n} k \\ = & - z (lim_{n \to \infty} \frac{\prod_{k = 1}^{n} k^{2}}{\prod_{k = 0}^{n} (k^{2} - z^{2})}) \dots (\prod_{k = m}^{n} A_{k}) (\prod_{k = m}^{n} B_{k}) = \prod_{k = m}^{n} A_{k} B_{k} \\ = & - z (lim_{n \to \infty} \frac{\prod_{k = 1}^{n} k^{2}}{(0^{2} - z^{2}) \prod_{k = 1}^{n} (k^{2} - z^{2})}) \\ = & - z \frac{1}{- z^{2}} (lim_{n \to \infty} \prod_{k = 1}^{n} \frac{k^{2}}{k^{2} - z^{2}}) \\ = & \frac{1}{z} \prod_{k = 1}^{\infty} \frac{k^{2}}{k^{2} - z^{2}} \\ = & \frac{1}{\frac{1}{\frac{1}{z} \prod_{k = 0}^{\infty} \frac{k^{2}}{k^{2} - z^{2}}}} \dots A = \frac{1}{\frac{1}{A}} \\ = & \frac{1}{z \prod_{k = 1}^{\infty} \frac{k^{2} - z^{2}}{k^{2}}} \\ = & \frac{π}{π} \frac{1}{z \prod_{k = 0}^{\infty} \frac{k^{2} - z^{2}}{k^{2}}} \\ = & \frac{π}{π z \prod_{k = 0}^{\infty} (1 - \frac{z^{2}}{k^{2}})} \\ = & \frac{π}{\sin (π z)} \\ \dots \sin (π z) = π z \prod_{k = 1}^{\infty} (1 - {(\frac{z}{k})}^{2}) (三 角 凾 数 の 無 限 乗 積 展 開 w i k i p e d i a) \end{array}

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