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2017年10月の1件の記事

三角関数の幾つかの2次無理式の積分

記事になるネタがないので、最近行った、幾つかの三角関数の無理式の積分計算をそのままアップする。

初めに、記号を決めておこう。

\begin{align*}
&0<k<1, 0<k^{\prime}<1, k^{2}+k^{\prime{2}}=1, 0\leq \varphi \leq \frac{\pi}{2}\\
&u(x) := k\sin(x), v(x) :=k\cos(x)\\
&\Delta(x,k) := \sqrt{1-k^{2}\sin^{2}x} = \sqrt{1-(u(x))^{2}} = \sqrt{k^{\prime2}+(v(x))^{2}}\\
&E(\varphi,k) := \int_{0}^{\varphi}\Delta({\theta},k)d\theta, \quad F(\varphi,k) := \int_{0}^{\varphi}\frac{d\theta}{\Delta({\theta},k)}\\
&I_{\mathrm s}^{n} := \int_{0}^{\varphi}\Delta({\theta},k)\sin^{n}(\theta)d\theta, \quad I_{\mathrm c}^{n} := \int_{0}^{\varphi}\Delta({\theta},k)\cos^{n}(\theta)d\theta\\
&J_{\mathrm s}^{n} := \int_{0}^{\varphi}\frac{\sin^{n}(\theta)}{\Delta({\theta},k)}d\theta, \quad J_{\mathrm c}^{n} := \int_{0}^{\varphi}\frac{\cos^{n}(\theta)}{\Delta({\theta},k)}d\theta\\
%&I_{\mathrm s}^{2} := \int_{0}^{\varphi}\Delta({\theta},k)\sin^{2}(\theta)d\theta, \quad I_{\mathrm c}^{2} := \int_{0}^{\varphi}\Delta({\theta},k)\cos^{2}(\theta)d\theta\\
\end{align*}

ここで、F(\varphi,k)E(\varphi,k) とが、それぞれ Legendre の第1種楕円積分と第2種楕円積分であるのは周知のとおり。そして、自明ながら、次の式が成り立っていることを注意しておく。

I_{\mathrm s}^{0} = I_{\mathrm c}^{0} = E(\varphi,k), \qquad J_{\mathrm s}^{0} = J_{\mathrm c}^{0} = F(\varphi,k)

I_{\mathrm s}^{2} + I_{\mathrm c}^{2} = E(\varphi,k), \qquad  J_{\mathrm s}^{2} + J_{\mathrm c}^{2} = F(\varphi,k)

以下では、I_{\mathrm s}^{n}, I_{\mathrm c}^{n}, J_{\mathrm s}^{n} 及び J_{\mathrm c}^{n} について、n=1,2 の場合の計算を行う。

下記の計算で用いる主な関係式は,次の通りの簡単なものだ。

\begin{align*}
 &\sin^{2}\theta = \frac{1 - (\Delta(\theta.k))^{2}}{k^2}\\
 &\cos^{2}\theta = 1 - \frac{1 - (\Delta(\theta.k))^{2}}{k^{2}} = \frac{-k^{\prime2} + (\Delta(\theta.k))^{2}}{k^{2}}\\
 &\odiff{\Delta}{\theta}(\theta,k) = -\frac{k^{2}\sin(\theta)\cos(\theta)}{\Delta(\theta,k)} = -\frac{u(\theta)v(\theta)}{\Delta(\theta,k)}\\
 &\odiff{u}{\theta}(\theta) = v(\theta), \quad \frac{du}{k} = \cos(\theta)d\theta\\
 &\odiff{v}{\theta}(\theta) = -u(\theta), \quad \frac{dv}{k} = -\sin(\theta)d\theta
\end{align*}

では、順次計算していこう。


\begin{align*}
 J^{1}_{\mathrm c} &= \int_{0}^{\varphi}\frac{\cos(\theta)}{\Delta(\theta,k)}d\theta\\
&= \int_{0}^{\varphi}\frac{\cos(\theta)}{\sqrt{1-(u(\theta))^{2}}}d\theta\\
&= \inverse{k}\int_{0}^{k\sin(\varphi)}\frac{du}{\sqrt{1-u^2}}\\
&= \inverse{k}\arcsin(k\sin(\varphi))\\
\end{align*}


\begin{align*}
 J^{1}_{\mathrm s} &= \int_{0}^{\varphi}\frac{\sin(\theta)}{\Delta(\theta,k)}d\theta\\
&= \int_{0}^{\varphi}\frac{\sin(\theta)}{\sqrt{k^{\prime2}+(v(\theta))^{2}}}d\theta\\
&= -\inverse{k}\int_{k}^{k\cos(\varphi)}\frac{v}{\sqrt{k^{\prime2}+v^{2}}}dv\\
&= -\inverse{k}\left[\ln(v+\sqrt{k^{\prime2}+v^{2}})\right]_{k}^{k\cos(\varphi)}\\
&= \inverse{k}\ln\left(\frac{k+\sqrt{k^{\prime2}+k^{2}}}{k\cos(\varphi)+\sqrt{k^{\prime2}+k^{2}\cos^{2}(\varphi)}}\right)\\
%&= \inverse{k}\ln\left(\frac{k+\sqrt{k^{\prime2}+k^{2}}}{k\cos(\varphi)+\sqrt{k^{\prime2}+k^{2}\cos^{2}(\varphi)}}\right)\\
&= \inverse{k}\ln\left(\frac{k+1}{k\cos(\varphi)+\sqrt{1-k^{2}\sin^{2}(\varphi)}}\right)\\
&= \inverse{k}\ln\left(\frac{(k+1)\left(k\cos(\varphi)-\sqrt{1-k^{2}\sin^{2}(\varphi)}\right)}{k^{2}\cos^{2}(\varphi)-(1-k^{2}\sin^{2}(\varphi))}\right)\\
&= \inverse{k}\ln\left(\frac{(k+1)\left(k\cos(\varphi)-\Delta(\varphi,k)\right)}{k^{2}-1}\right)\\
&= \inverse{k}\ln\left(\frac{\Delta(\varphi,k)-k\cos(\varphi)}{1-k}\right)\\
\end{align*}


\begin{align*}
 I^{1}_{\mathrm c} &= \int_{0}^{\varphi}\Delta(\theta,k)\cos(\theta)d\theta\\
&= \inverse{k}\int_{0}^{k\sin(\varphi)}\sqrt{1-u^{2}}du\\
&= \inverse{k}\left[\inverse{2}\left(u\sqrt{1-u^{2}}+\arcsin(u)\right)\right]_{0}^{k\sin(\varphi)}\\
&= \inverse{2k}\left(k\sin(\varphi)\sqrt{1-k^{2}\sin^{2}(\varphi)} + \arcsin(k\sin(\varphi))\right)\\
&= \frac{\sin(\varphi)\Delta(\varphi,k)}{2} + \inverse{2k}\arcsin(k\sin(\varphi))
\end{align*}


\begin{align*}
 I^{1}_{\mathrm s} &= \int_{0}^{\varphi}\Delta(\theta,k)\sin(\theta)d\theta\\
&= -\inverse{k}\int_{k}^{k\cos(\varphi)}\sqrt{k^{\prime2}+v^{2}}dv\\
&= -\inverse{k}\left\{\inverse{2}\left[v\sqrt{k^{\prime2}+v^{2}} + k^{\prime2}\ln(v+\sqrt{k^{\prime2}+v^{2}})\right]_{k}^{k\cos(\varphi)}\right\}\\
&= -\inverse{2k}\left\{k\cos(\varphi)\sqrt{k^{\prime2}+k^{2}\cos^{2}(\varphi)} - k\sqrt{k^{\prime2}+k^{2}}\right\}\\
&\qquad\qquad\qquad - \frac{k^{\prime2}}{2k}\ln\left\{\frac{k\cos(\varphi) + \sqrt{k^{\prime2}+k^{2}\cos^{2}(\varphi)}}{k+\sqrt{k^{\prime2}+k^{2}}}\right\}\\
&= -\inverse{2}\cos(\varphi)\Delta(\varphi,k) + \inverse{2} - \frac{1-k^{2}}{2k}\ln\left\{\frac{k\cos(\varphi)+\Delta(\varphi,k)}{k+1}\right\}
\end{align*}


\begin{align*}
 J^{2}_{\mathrm s} &= \int_{0}^{\varphi}\frac{\sin^{2}(\theta)}{\Delta(\theta,k)}d\theta\\
&= \int_{0}^{\varphi}\inverse{\Delta(\theta,k)}\left\{\inverse{k^{2}}(1-(1-k^{2}\sin^{2}(\theta))\right\}d\theta\\
&= \inverse{k^{2}}\int_{0}^{\varphi}\left(\inverse{\Delta(\theta,k)} - \Delta(\theta,k)\right)d\theta\\
&= \inverse{k^{2}}\left(F(\varphi,k)-E(\varphi,k)\right)
\end{align*}


\begin{align*}
 J^{2}_{\mathrm c} &= F(\varphi,k) - J^{2}_{\mathrm s}\\
&=(1-\inverse{k^{2}})F(\varphi,k) + \inverse{k^{2}}E(\varphi,k)\\
&=\inverse{k^{2}}E(\varphi,k) - \frac{1-k^{2}}{k^{2}}F(\varphi,k)
\end{align*}


\begin{align*}
 I^{2}_{\mathrm c} &= \int_{0}^{\varphi}\Delta(\theta,k)\cos^{2}(\theta)d\theta = \int_{0}^{\varphi}\Delta(\theta,k)\left(\frac{1+\cos(2\theta)}{2}\right)d\theta\\
&=\inverse{2}\int_{0}^{\varphi}\Delta(\theta,k)d\theta + \inverse{2}\int_{0}^{\varphi}\Delta(\theta,k)\cos(2\theta)d\theta\\
&=\inverse{2}E(\varphi,k) + \inverse{2}\int_{0}^{\varphi}\Delta(\theta,k)\left(\inverse{2}\sin(2\theta)\right)^{\prime}d\theta\\
&=\inverse{2}E(\varphi,k) + \inverse{4}\left\{\left[\Delta(\theta,k)\sin(2\theta)\right]_{0}^{\varphi} - \int_{0}^{\varphi}
\left(\Delta(\theta,k)\right)^{\prime}\sin(2\theta)d\theta\right\}\\
&=\inverse{2}E(\varphi,k) + \inverse{4}\left[\Delta(\theta,k)\sin(2\theta)\right]_{0}^{\varphi}\\
&\qquad\qquad\qquad\qquad + \inverse{4}\int_{0}^{\varphi}\frac{k^{2}\sin(\theta)\cos(\theta)}{\Delta(\theta,k)}\sin(2\theta)d\theta\\
&=\inverse{2}E(\varphi,k) + \inverse{4}\Delta(\varphi,k)\sin(2\varphi)\\
&\qquad\qquad\qquad\qquad + \frac{k^{2}}{2}\int_{0}^{\varphi}\frac{\sin^2(\theta)\cos^2(\theta)}{\Delta(\theta,k)}d\theta\\
&=\inverse{2}E(\varphi,k) + \inverse{2}\Delta(\varphi,k)\sin(\varphi)\cos(\varphi)\\
&\qquad\qquad\qquad\qquad + \frac{k^{2}}{2}\int_{0}^{\varphi}\frac{\sin^2(\theta)\cos^2(\theta)}{\Delta(\theta,k)}d\theta
\end{align*}

しかるに
\begin{align*}
 &\frac{k^{2}}{2}\int_{0}^{\varphi}\frac{\sin^2(\theta)\cos^2(\theta)}{\Delta(\theta,k)}d\theta\\
&\quad = \frac{k^{2}}{2}\int_{0}^{\varphi}\left(\inverse{{\Delta(\theta,k)}}\right)\left(\frac{1 - (\Delta(x.k))^{2}}{k^2}\right)\left(\frac{-k^{\prime2} + (\Delta(x.k))^{2}}{k^{2}}\right)d\theta\\
&\quad = \inverse{2k^{2}}\int_{0}^{\varphi}\left(\inverse{\Delta(\theta,k)}\right)\left(-k^{\prime2} + (1+k^{\prime2})(\Delta(x.k))^{2} - (\Delta(x.k))^{4}\right)d\theta\\
&\quad = -\frac{k^{\prime2}}{2k^{2}}\int_{0}^{\varphi}\frac{d\theta}{\Delta(\theta,k)} + \frac{1+k^{\prime2}}{2k^{2}}\int_{0}^{\varphi}\Delta(x.k)d\theta\\
&\qquad\qquad\qquad\qquad - \inverse{2k^{2}}\int_{0}^{\varphi}(1-k^{2}\sin^{2}\theta)\Delta(x.k)d\theta\\
&\quad = -\frac{k^{\prime2}}{2k^{2}}F(\varphi,k) + \frac{k^{\prime2}}{2k^{2}}E(\varphi,k) + \inverse{2}\int_{0}^{\varphi}\sin^{2}\theta\Delta(x.k)d\theta\\
&\quad = -\frac{k^{\prime2}}{2k^{2}}F(\varphi,k) + \frac{k^{\prime2}}{2k^{2}}E(\varphi,k) + \inverse{2}\int_{0}^{\varphi}(1 - \cos^{2}\theta)\Delta(x.k)d\theta\\
&\quad = -\frac{k^{\prime2}}{2k^{2}}F(\varphi,k) + \left(\frac{k^{\prime2}}{2k^{2}} + \inverse{2}\right)E(\varphi,k) - \inverse{2}I^{2}_{\mathrm c}\\
\end{align*}
だから、まとめると
\begin{align*}
\frac{3}{2}I^{2}_{\mathrm c} &= \inverse{2}E(\varphi,k) + \inverse{2}\Delta(\varphi,k)\sin(\varphi)\cos(\varphi)\\
&\qquad\qquad + \left(-\frac{k^{\prime2}}{2k^{2}}F(\varphi,k) + \left(\frac{k^{\prime2}}{2k^{2}} + \inverse{2}\right)E(\varphi,k)\right)\\
&= \inverse{2}\Delta(\varphi,k)\sin(\varphi)\cos(\varphi) + \left(\frac{k^{\prime2}+2k^{2}}{2k^{2}}\right)E(\varphi,k) - \frac{k^{\prime2}}{2k^{2}}F(\varphi,k)\\
&= \inverse{2}\Delta(\varphi,k)\sin(\varphi)\cos(\varphi) + \frac{1+k^{2}}{2k^{2}}E(\varphi,k) - \frac{1-k^{2}}{2k^{2}}F(\varphi,k)
\end{align*}
となる。

結局
\begin{align*}
 I^{2}_{\mathrm c} = \inverse{3}\Delta(\varphi,k)\sin(\varphi)\cos(\varphi) + \frac{1+k^{2}}{3k^{2}}E(\varphi,k) - \frac{1-k^{2}}{3k^{2}}F(\varphi,k)
\end{align*}
が得られる。


\begin{align*}
 I^{2}_{\mathrm s} &= E(\varphi,k) - I^{2}_{\mathrm c}\\
 &= -\inverse{3}\Delta(\varphi,k)\sin(\varphi)\cos(\varphi) + \frac{2k^{2}-1}{3k^{2}}E(\varphi,k) + \frac{1-k^{2}}{3k^{2}}F(\varphi,k)
\end{align*}\end{align*}


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