K²+K'² を分母にもつ関数の定積分

523

この著者は初心者として投稿しています。間違いや考慮が足りていない点が含まれている可能性が高いです。見つけたらコメント欄で優しく指摘してあげましょう。

定義・記法

以下，当記事での定義および記法を示します．

\begin{aligned} β_{r}^{} & = 2^{- 2 r} (\binom{2 r}{r}) \\ K (x) & = \frac{π}{2} \sum_{n = 0}^{\infty} β_{n}^{2} x^{2 n} \\ K^{'} (x) & = K (\sqrt{1 - x^{2}}) \\ \frac{1}{\sqrt{1 - 2 t x + t^{2}}} & = \sum_{n = 0}^{\infty} t^{n} P_{n}^{} (x) \\ \frac{1}{1 - 2 t x + t^{2}} & = \sum_{n = 0}^{\infty} t^{n} U_{n}^{} (x) \\ G & = \sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{(2 n + 1)^{2}} \end{aligned}

また，分子の $K (x), K^{'} (x)$ の数に応じて，積分の"重み"を決定します．すなわち ${K (x)}^{a} {K^{'} (x)}^{b}$ の場合を重み $a + b - 1$ とします．

使用アプリケーション

$W o l f r a m A l p h a$
PARI/GP

(数値実験１)　重み $0$

$x^{n}$

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} x^{0} d x & = \frac{1}{4} \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} x^{2} d x & = \frac{5}{64} \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} x^{4} d x & = \frac{11}{256} \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} x^{6} d x & = \frac{469}{16384} \end{aligned}

$P_{n}^{} (x)$

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} P_{0}^{} (x) d x & = \frac{1}{4} \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} P_{2}^{} (x) d x & = - \frac{1}{128} \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} P_{4}^{} (x) d x & = - \frac{23}{2048} \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} P_{6}^{} (x) d x & = \frac{499}{262144} \end{aligned}

$U_{n}^{} (x)$

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{0}^{} (x) d x & = \frac{1}{4} \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{4}^{} (x) d x & = 0 \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{8}^{} (x) d x & = 0 \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{12}^{} (x) d x & = 0 \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{2}^{} (x) d x & = \frac{1}{16} \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{6}^{} (x) d x & = - \frac{5}{256} \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{10}^{} (x) d x & = - \frac{11}{1024} \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{14}^{} (x) d x & = \frac{469}{65536} \end{aligned}

★

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{4 n}^{} (x) d x & = \frac{1}{4} δ_{0, n}^{} \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} x^{4 n} d x & = 4 (- 1)^{n} \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{8 n + 2}^{} (x) d x \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} x^{4 n + 2} d x & = 4 (- 1)^{n + 1} \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{8 n + 6}^{} (x) d x \\ \int_{0}^{1} \frac{K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} x^{2 n} d x & = [x^{2 n}] \frac{1}{x^{2}} (1 - \frac{π}{2} \frac{1}{K (x)}) \end{aligned}

(数値実験２)　重み $1$

$x^{n}$

\begin{aligned} \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} x^{1} d x & = \frac{1}{4} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} x^{3} d x & = \frac{19}{128} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} x^{5} d x & = \frac{41}{384} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} x^{7} d x & = \frac{5491}{65536} \end{aligned}

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} x^{1} d x & = \frac{1}{4} \\ \int_{0}^{1} \frac{K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} x^{3} d x & = \frac{13}{128} \\ \int_{0}^{1} \frac{K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} x^{5} d x & = \frac{23}{384} \\ \int_{0}^{1} \frac{K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} x^{7} d x & = \frac{2701}{65536} \end{aligned}

$P_{n}^{} (x)$

\begin{aligned} \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} P_{1}^{} (x) d x & = \frac{1}{4} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} P_{3}^{} (x) d x & = - \frac{31}{256} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} P_{5}^{} (x) d x & = \frac{53}{1024} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} P_{7}^{} (x) d x & = - \frac{38327}{1048576} \end{aligned}

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} P_{1}^{} (x) d x & = \frac{1}{4} \\ \int_{0}^{1} \frac{K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} P_{3}^{} (x) d x & = - \frac{1}{256} \\ \int_{0}^{1} \frac{K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} P_{5}^{} (x) d x & = \frac{11}{1024} \\ \int_{0}^{1} \frac{K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} P_{7}^{} (x) d x & = - \frac{2633}{1048576} \end{aligned}

$U_{n}^{} (x)$

\begin{aligned} \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{1}^{} (x) d x & = \frac{1}{2} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{5}^{} (x) d x & = \frac{1}{6} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{9}^{} (x) d x & = \frac{1}{10} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{13}^{} (x) d x & = \frac{1}{14} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{3}^{} (x) d x & = \frac{3}{16} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{7}^{} (x) d x & = \frac{51}{512} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{11}^{} (x) d x & = \frac{35}{512} \\ \int_{0}^{1} \frac{K (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{15}^{} (x) d x & = \frac{13683}{262144} \end{aligned}

★

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} (x^{2 n + 1} - 4 U_{4 n + 3}^{} (x)) d x & = \frac{1}{n + 1} \\ \int_{0}^{1} \frac{K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} \frac{x}{1 - z^{2} x^{2}} d x & = \frac{1}{z^{2}} (\ln \frac{4}{z} - \frac{π}{2} \frac{K^{'} (z)}{K (z)}) \end{aligned}

(数値実験３)　重み $2$

$x^{n}$

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} x^{0} d x & = β_{0}^{2} ζ (2) \\ \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} x^{2} d x & = β_{1}^{2} ζ (2) - \frac{1}{8} \\ \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} x^{4} d x & = β_{2}^{2} ζ (2) - \frac{49}{512} \\ \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} x^{6} d x & = β_{3}^{2} ζ (2) - \frac{1405}{18432} \end{aligned}

$P_{n}^{} (x)$

\begin{aligned} \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} P_{0}^{} (x) d x & = \frac{1}{2} β_{0}^{3} ζ (2) \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} P_{2}^{} (x) d x & = \frac{3}{16} - \frac{1}{2} β_{1}^{3} ζ (2) \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} P_{4}^{} (x) d x & = \frac{1}{2} β_{2}^{3} ζ (2) - \frac{205}{4096} \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} P_{6}^{} (x) d x & = \frac{3605}{98304} - \frac{1}{2} β_{3}^{3} ζ (2) \end{aligned}

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} P_{0}^{} (x) d x & = β_{0}^{3} ζ (2) \\ \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} P_{2}^{} (x) d x & = - (β_{1}^{3} ζ (2) + \frac{3}{16}) \\ \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} P_{4}^{} (x) d x & = β_{2}^{3} ζ (2) + \frac{205}{4096} \\ \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} P_{6}^{} (x) d x & = - (β_{3}^{3} ζ (2) + \frac{3605}{98304}) \end{aligned}

$U_{n}^{} (x)$

\begin{aligned} \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{0}^{} (x) d x & = \frac{1}{2} β_{0}^{2} ζ (2) \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{4}^{} (x) d x & = \frac{1}{2} β_{1}^{2} ζ (2) + \frac{1}{32} \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{8}^{} (x) d x & = \frac{1}{2} β_{2}^{2} ζ (2) + \frac{49}{2048} \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{12}^{} (x) d x & = \frac{1}{2} β_{3}^{2} ζ (2) + \frac{1405}{73728} \\ \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} U_{0}^{} (x) d x & = β_{0}^{2} ζ (2) \\ \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} U_{4}^{} (x) d x & = β_{1}^{2} ζ (2) - \frac{1}{32} \\ \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} U_{8}^{} (x) d x & = β_{2}^{2} ζ (2) - \frac{49}{2048} \\ \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} U_{12}^{} (x) d x & = β_{3}^{2} ζ (2) - \frac{1405}{73728} \end{aligned}

\begin{aligned} \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{2}^{} (x) d x & = \frac{1}{2} \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{6}^{} (x) d x & = \frac{2}{9} \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{10}^{} (x) d x & = \frac{32}{225} \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{14}^{} (x) d x & = \frac{128}{1225} \end{aligned}

★

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)^{3}}{K (x)^{2} + K^{'} (x)^{2}} (4 U_{4 n}^{} (x) - x^{2 n}) d x & = \frac{π^{2}}{2} β_{n}^{2} \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} U_{4 n + 2}^{} (x) d x & = \frac{2}{(2 n + 1)^{2} β_{n}^{2}} \end{aligned}

(数値実験４)　重み $3$

$U_{n}^{} (x)$

\begin{aligned} \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{1}^{} (x) d x & = \frac{3}{2} ζ (3) \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{5}^{} (x) d x & = \frac{3}{2} \frac{1}{2} ζ (3) - \frac{3}{16} \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{9}^{} (x) d x & = \frac{3}{2} \frac{11}{32} ζ (3) - \frac{363}{2048} \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} U_{13}^{} (x) d x & = \frac{3}{2} \frac{17}{64} ζ (3) - \frac{5833}{36864} \end{aligned}

\begin{aligned} \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} P_{1}^{} (x) d x & = \frac{3}{4} ζ (3) \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} P_{3}^{} (x) d x & = - \frac{3}{4} \frac{1}{4} ζ (3) \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} P_{5}^{} (x) d x & = \frac{3}{4} \frac{53}{256} ζ (3) - \frac{189}{4096} \end{aligned}

その他

重み $0$

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)^{4} K (x)}{{(K (x)^{2} + K^{'} (x)^{2})}^{2}} d x & = \frac{9}{16} ζ (3) \\ \int_{0}^{1} \frac{K^{'} (x)^{6} K (x)}{{(K (x)^{2} + K^{'} (x)^{2})}^{3}} d x & = \frac{45}{64} ζ (3) - \frac{93}{256} ζ (5) \\ \int_{0}^{1} \frac{K^{'} (x)^{8} K (x)}{{(K (x)^{2} + K^{'} (x)^{2})}^{4}} d x & = \frac{105}{128} ζ (3) - \frac{159}{256} ζ (5) \end{aligned}

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)^{4}}{K (x) (K (x)^{2} + K^{'} (x)^{2})} d x & = \frac{21}{8} ζ (3) \\ \int_{0}^{1} \frac{K^{'} (x)^{6}}{K (x) {(K (x)^{2} + K^{'} (x)^{2})}^{2}} d x & = \frac{33}{16} ζ (3) \\ \int_{0}^{1} \frac{K^{'} (x)^{8}}{K (x) {(K (x)^{2} + K^{'} (x)^{2})}^{3}} d x & = \frac{87}{64} ζ (3) + \frac{93}{256} ζ (5) \\ \int_{0}^{1} \frac{K^{'} (x)^{10}}{K (x) {(K (x)^{2} + K^{'} (x)^{2})}^{4}} d x & = \frac{69}{128} ζ (3) + \frac{63}{64} ζ (5) \\ \int_{0}^{1} \frac{K^{'} (x)^{12}}{K (x) {(K (x)^{2} + K^{'} (x)^{2})}^{5}} d x & = - \frac{393}{1024} ζ (3) + \frac{8547}{4096} ζ (5) - \frac{9525}{32768} ζ (7) \end{aligned}

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)^{4}}{K (x)^{3}} d x & = \frac{93}{4} ζ (5) \\ \int_{0}^{1} \frac{K^{'} (x)^{6}}{K (x)^{3} (K (x)^{2} + K^{'} (x)^{2})} d x & = - \frac{21}{8} ζ (3) + \frac{93}{4} ζ (5) \end{aligned}

重み $3$

\begin{aligned} \int_{0}^{1} \frac{3 K (x)^{4} - 4 K (x)^{2} K^{'} (x)^{2}}{K (x)^{2} + K^{'} (x)^{2}} x d x & = 0 \end{aligned}

重み $4$

\begin{aligned} \int_{0}^{1} \frac{K (x)^{5} + K (x)^{2} K^{'} (x)^{3} - K (x) K^{'} (x)^{4}}{K (x)^{2} + K^{'} (x)^{2}} d x & = 0 \\ \int_{0}^{1} \frac{K (x)^{4} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} d x - \int_{0}^{1} \frac{K (x)^{5} - K (x)^{4} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} x d x - \int_{0}^{1} \frac{K (x)^{3} K^{'} (x)^{2} - K (x) K^{'} (x)^{4}}{K (x)^{2} + K^{'} (x)^{2}} x^{2} d x & = 0 \\ \int_{0}^{1} \frac{K (x)^{5} + 2 K (x)^{3} K^{'} (x)^{2} + K (x) K^{'} (x)^{4} - K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x & = 0 \\ \int_{0}^{1} \frac{12 K (x)^{5} - 2 K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x - \int_{0}^{1} \frac{3 K (x)^{4} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} x d x & = 0 \\ \int_{0}^{1} \frac{32 (x)^{5} - 5 K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x - \int_{0}^{1} \frac{8 K (x)^{4} K^{'} (x)}{K (x)^{2} + K^{'} (x)^{2}} x d x & = 0 \\ \int_{0}^{1} \frac{11 K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x - \int_{0}^{1} \frac{96 K (x)^{4} K^{'} (x) + 39 K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} x^{2} d x & = 0 \end{aligned}

\begin{aligned} \int_{0}^{1} \frac{K^{'} (x)^{6}}{K (x) (K (x)^{2} + K^{'} (x)^{2})} d x & = \frac{Γ {(\frac{1}{4})}^{8}}{160 π^{2}} \end{aligned}

$λ K (x)^{2} + μ K^{'} (x)^{2}$

重み $2$

\begin{aligned} \int_{0}^{1} \frac{K (x)^{3}}{K (x)^{2} + \frac{1}{2^{2}} K^{'} (x)^{2}} d x & = 6 G - \frac{7}{2} ζ (3) \\ \int_{0}^{1} \frac{K (x) K^{'} (x)^{2}}{K (x)^{2} + \frac{1}{2^{2}} K^{'} (x)^{2}} d x & = - 16 G + 14 ζ (3) \end{aligned}

\begin{aligned} \int_{0}^{1} \frac{K (x) K^{'} (x)^{2}}{K (x)^{2} + \frac{1}{3^{2}} K^{'} (x)^{2}} d x & = \frac{19}{8} ζ (3) \\ \int_{0}^{1} \frac{K (x) K^{'} (x)^{2}}{\frac{1}{3^{2}} K (x)^{2} + K^{'} (x)^{2}} d x & = \frac{11}{8} ζ (3) \end{aligned}

重み $4$

\begin{aligned} \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + \frac{1}{2^{2}} K^{'} (x)^{2}} d x & = \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x + \frac{3}{32} \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \\ \int_{0}^{1} \frac{K (x)^{4} K^{'} (x)}{K (x)^{2} + \frac{1}{2^{2}} K^{'} (x)^{2}} d x & = \frac{5}{2} \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x - \frac{5}{24} \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \\ \int_{0}^{1} \frac{K (x)^{3} K^{'} (x)^{2}}{K (x)^{2} + \frac{1}{2^{2}} K^{'} (x)^{2}} d x & = - 4 \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x + \frac{9}{8} \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \\ \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)^{3}}{K (x)^{2} + \frac{1}{2^{2}} K^{'} (x)^{2}} d x & = - 10 \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x + \frac{5}{2} \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \\ \int_{0}^{1} \frac{K (x) K^{'} (x)^{4}}{K (x)^{2} + \frac{1}{2^{2}} K^{'} (x)^{2}} d x & = 16 \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x - 2 \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \\ \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + \frac{1}{2^{2}} K^{'} (x)^{2}} d x & = 40 \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x - 5 \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \end{aligned}

\begin{aligned} 6 \int_{0}^{1} \frac{K (x)^{5}}{\frac{1}{2^{2}} K (x)^{2} + K^{'} (x)^{2}} d x + 12 \int_{0}^{1} \frac{K (x)^{4} K^{'} (x)}{\frac{1}{2^{2}} K (x)^{2} + K^{'} (x)^{2}} d x & = - 1536 \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x + 305 \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \\ 2 \int_{0}^{1} \frac{K (x)^{3} K^{'} (x)^{2}}{\frac{1}{2^{2}} K (x)^{2} + K^{'} (x)^{2}} d x + 4 \int_{0}^{1} \frac{K (x)^{2} K^{'} (x)^{3}}{\frac{1}{2^{2}} K (x)^{2} + K^{'} (x)^{2}} d x & = 128 \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x - 23 \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \\ \int_{0}^{1} \frac{K (x) K^{'} (x)^{4}}{\frac{1}{2^{2}} K (x)^{2} + K^{'} (x)^{2}} d x + 2 \int_{0}^{1} \frac{K^{'} (x)^{5}}{\frac{1}{2^{2}} K (x)^{2} + K^{'} (x)^{2}} d x & = - 16 \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x + 6 \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \end{aligned}

\begin{aligned} 729 \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + \frac{1}{3^{2}} K^{'} (x)^{2}} d x & = - 731 \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x + 380 \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \\ 648 \int_{0}^{1} \frac{K (x)^{3} K^{'} (x)^{2}}{K (x)^{2} + \frac{1}{3^{2}} K^{'} (x)^{2}} d x & = 5848 \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x - 853 \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \\ 36 \int_{0}^{1} \frac{K (x) K^{'} (x)^{4}}{K (x)^{2} + \frac{1}{3^{2}} K^{'} (x)^{2}} d x & = - 2924 \int_{0}^{1} \frac{K (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x + 629 \int_{0}^{1} \frac{K^{'} (x)^{5}}{K (x)^{2} + K^{'} (x)^{2}} d x \end{aligned}

\begin{array}{r}  \end{array}

投稿日：2024年1月21日

更新日：2024年5月24日

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K²+K'² を分母にもつ関数の定積分

定義・記法
使用アプリケーション
(数値実験１)　重み$0$
$x^n$
$P_n^{}(x)$
$U_n^{}(x)$
(数値実験２)　重み$1$
$x^n$
$P_n^{}(x)$
$U_n^{}(x)$
(数値実験３)　重み$2$
$x^n$
$P_n^{}(x)$
$U_n^{}(x)$
(数値実験４)　重み$3$
$U_n^{}(x)$
その他
重み$0$
重み$3$
重み$4$
$\D\lambda K(x)^2+ \mu K'(x)^2$
重み$2$
重み$4$

K²+K'² を分母にもつ関数の定積分

定義・記法

使用アプリケーション

(数値実験 １) 重み0

xn

Pn(x)

Un(x)

(数値実験 ２) 重み1

xn

Pn(x)

Un(x)

(数値実験 ３) 重み2

xn

Pn(x)

Un(x)

(数値実験 ４) 重み3

Un(x)

その他

重み0

重み3

重み4

λK(x)2+μK′(x)2

重み2

重み4

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(数値実験１)　重み $0$

$x^{n}$

$P_{n}^{} (x)$

$U_{n}^{} (x)$

(数値実験２)　重み $1$

$x^{n}$

$P_{n}^{} (x)$

$U_{n}^{} (x)$

(数値実験３)　重み $2$

$x^{n}$

$P_{n}^{} (x)$

$U_{n}^{} (x)$

(数値実験４)　重み $3$

$U_{n}^{} (x)$

重み $0$

重み $3$

重み $4$

$λ K (x)^{2} + μ K^{'} (x)^{2}$

重み $2$

重み $4$