\(\lim_{x\rightarrow \frac{\pi}{2}}\ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}\)を求める
高階の微分を求めておく
一階から順に求めておく．
$$\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d}x} \ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
&=&\frac{1}{\tan{\left(\frac{x}{2}\right)}}\left(\frac{\mathrm{d}}{\mathrm{d}x}\tan{\left(\frac{x}{2}\right)}\right)
\;\cdots\;u=\tan{\left(\frac{x}{2}\right)},f=\ln{\left(u\right)},\frac{\mathrm{d}f}{\mathrm{d}x}=\frac{\mathrm{d}f}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{1}{u}\frac{\mathrm{d}u}{\mathrm{d}x}
\\&=&\frac{1}{
\frac{\sin{\left(\frac{x}{2}\right)}}{\cos{\left(\frac{x}{2}\right)}}
}
\left(\frac{\mathrm{d}}{\mathrm{d}x}
\frac{\sin{\left(\frac{x}{2}\right)}}{\cos{\left(\frac{x}{2}\right)}}
\right)
\\&=&\frac{\cos{\left(\frac{x}{2}\right)}}{\sin{\left(\frac{x}{2}\right)}}
\left(\frac{\mathrm{d}}{\mathrm{d}x}
\sin{\left(\frac{x}{2}\right)}\cos^{-1}{\left(\frac{x}{2}\right)}
\right)
\\&=&\frac{\cos{\left(\frac{x}{2}\right)}}{\sin{\left(\frac{x}{2}\right)}}
\left\{
\left(\frac{\mathrm{d}}{\mathrm{d}x}\sin{\left(\frac{x}{2}\right)}\right)\cos^{-1}{\left(\frac{x}{2}\right)}
+\sin{\left(\frac{x}{2}\right)}\left(\frac{\mathrm{d}}{\mathrm{d}x}\cos^{-1}{\left(\frac{x}{2}\right)}\right)
\right\}
\\&=&\frac{\cos{\left(\frac{x}{2}\right)}}{\sin{\left(\frac{x}{2}\right)}}
\left[
\left(\cancel{\cos{\left(\frac{x}{2}\right)}}\cdot\frac{1}{2}\right)\cancel{\cos^{-1}{\left(\frac{x}{2}\right)}}
+\sin{\left(\frac{x}{2}\right)}\left\{-\cos^{-2}{\left(\frac{x}{2}\right)}\left(-\sin{\left(\frac{x}{2}\right)}\cdot\frac{1}{2}\right)\right\}
\right]
\\&=&\frac{\cos{\left(\frac{x}{2}\right)}}{\sin{\left(\frac{x}{2}\right)}}
\cdot\frac{1}{2}\left\{
1+\sin^2{\left(\frac{x}{2}\right)}\cos^{-2}{\left(\frac{x}{2}\right)}
\right\}
\\&=&\frac{1}{2}\left(
\frac{\cos{\left(\frac{x}{2}\right)}}{\sin{\left(\frac{x}{2}\right)}}
+\frac{\cancel{\cos{\left(\frac{x}{2}\right)}}}{\cancel{\sin{\left(\frac{x}{2}\right)}}}\sin^{\cancel{2}1}{\left(\frac{x}{2}\right)}\cos^{\cancel{-2}-1}{\left(\frac{x}{2}\right)}
\right)
\\&=&\frac{1}{2}\left(
\frac{\cos{\left(\frac{x}{2}\right)}}{\sin{\left(\frac{x}{2}\right)}}
+\frac{\sin{\left(\frac{x}{2}\right)}}{\cos{\left(\frac{x}{2}\right)}}
\right)
\\&=&\frac{1}{2}
\frac{
\cos^2{\left(\frac{x}{2}\right)}+\sin^2{\left(\frac{x}{2}\right)}
}{\sin{\left(\frac{x}{2}\right)}\cos{\left(\frac{x}{2}\right)}}
\\&=&
\frac{1}{2\sin{\left(\frac{x}{2}\right)}\cos{\left(\frac{x}{2}\right)}}
\\&=&\frac{1}{\sin{(x)}}\;\cdots\;\sin{(x)}=2\sin{\left(\frac{x}{2}\right)}\cos{\left(\frac{x}{2}\right)}
\\\;
\\\frac{\mathrm{d}^2}{\mathrm{d}x^2} \ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
&=&\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{\sin{(x)}}\right)
\\&=&-\frac{1}{\sin^2{(x)}}\left\{\frac{\mathrm{d}}{\mathrm{d}x}\sin{(x)}\right\}
\\&=&-\frac{1}{\sin^2{(x)}}\cos{(x)}
\\&=&-\frac{\cos{(x)}}{\sin^2{(x)}}
\\\;
\\\frac{\mathrm{d}^3}{\mathrm{d}x^3} \ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
&=&\frac{\mathrm{d}}{\mathrm{d}x}\left(-\frac{\cos{(x)}}{\sin^2{(x)}}\right)
\\&=&-\frac{\mathrm{d}}{\mathrm{d}x}\cos{(x)}\sin^{-2}{(x)}
\\&=&-\left\{
\left( \frac{\mathrm{d}}{\mathrm{d}x}\cos{(x)}\right) \sin^{-2}{(x)}
+ \cos{(x)}\left(\frac{\mathrm{d}}{\mathrm{d}x}\sin^{-2}{(x)}\right)
\right\}
\\&=&-\left\{
\left( -\sin{(x)}\right) \sin^{-2}{(x)}
+ \cos{(x)}\left(-2\sin^{-3}{(x)}\cos{(x)}\right)
\right\}
\\&=&-\left\{
-\sin^{-1}{(x)}-2\sin^{-3}{(x)}\cos^2{(x)}
\right\}
\\&=&\sin^{-1}{(x)}+2\sin^{-3}{(x)}\cos^2{(x)}
\\&=&\frac{1}{\sin{(x)}}+\frac{2\cos^2{(x)}}{\sin^3{(x)}}
\\\;
\\\frac{\mathrm{d}^4}{\mathrm{d}x^4} \ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
&=&\frac{\mathrm{d}}{\mathrm{d}x}\left(
\frac{1}{\sin{(x)}}+\frac{2\cos^2{(x)}}{\sin^3{(x)}}
\right)
\\&=&\frac{\mathrm{d}}{\mathrm{d}x}\sin^{-1}{(x)}
+2\frac{\mathrm{d}}{\mathrm{d}x}\cos^2{(x)}\sin^{-3}{(x)}
\\&=&
-\sin^{-2}{(x)}\cos{(x)}
+2\left\{
\left(\frac{\mathrm{d}}{\mathrm{d}x}\cos^2{(x)}\right)\sin^{-3}{(x)}
+\cos^2{(x)}\left(\frac{\mathrm{d}}{\mathrm{d}x}\sin^{-3}{(x)}\right)
\right\}
\\&=&
-\sin^{-2}{(x)}\cos{(x)}
+2\left[
\left\{2\cos{(x)}\left(-\sin{(x)}\right)\right\}\sin^{-3}{(x)}
+\cos^2{(x)}\left(-3\sin^{-4}{(x)}\cos{(x)}\right)
\right]
\\&=&-\frac{\cos{(x)}}{\sin^{2}{(x)}}
+2\left[
-2\frac{\cos{(x)}}{\sin^{2}{(x)}}
-3\frac{\cos^3{(x)}}{\sin^{4}{(x)}}
\right]
\\&=&-\frac{\cos{(x)}}{\sin^{2}{(x)}}
-4\frac{\cos{(x)}}{\sin^{2}{(x)}}
-6\frac{\cos^3{(x)}}{\sin^{4}{(x)}}
\\&=&-5\frac{\cos{(x)}}{\sin^{2}{(x)}}
-6\frac{\cos^3{(x)}}{\sin^{4}{(x)}}
\\\;
\\\frac{\mathrm{d}^5}{\mathrm{d}x^5} \ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
&=&\frac{\mathrm{d}}{\mathrm{d}x}\left(
-5\frac{\cos{(x)}}{\sin^{2}{(x)}}
-6\frac{\cos^3{(x)}}{\sin^{4}{(x)}}
\right)
\\&=&-5\frac{\mathrm{d}}{\mathrm{d}x}\cos{(x)}\sin^{-2}{(x)}
-6\frac{\mathrm{d}}{\mathrm{d}x}\cos^3{(x)}\sin^{-4}{(x)}
\\&=&-5\left\{
\left(\frac{\mathrm{d}}{\mathrm{d}x}\cos{(x)}\right)\sin^{-2}{(x)}
+\cos{(x)}\frac{\mathrm{d}}{\mathrm{d}x}\sin^{-2}{(x)}
\right\}
-6\left\{
\left(\frac{\mathrm{d}}{\mathrm{d}x}\cos^3{(x)}\right)\sin^{-4}{(x)}
+\cos^3{(x)}\frac{\mathrm{d}}{\mathrm{d}x}\sin^{-4}{(x)}
\right\}
\\&=&-5\left\{
\left(-\sin{(x)}\right)\sin^{-2}{(x)}
+\cos{(x)}\left(-2\sin^{-3}{(x)}\cos{(x)}\right)
\right\}
-6\left\{
\left(-3\cos^2{(x)}\sin{(x)}\right)\sin^{-4}{(x)}
+\cos^3{(x)}\left(-4\sin^{-5}{(x)}\cos{(x)}\right)
\right\}
\\&=&5\frac{1}{\sin{(x)}}
+28\frac{\cos^2{(x)}}{\sin^{3}{(x)}}
+24\frac{\cos^4{(x)}}{\sin^{5}{(x)}}
\end{eqnarray}$$
\(x=\frac{\pi}{2}\)でのテーラー展開を求めておく
$$\begin{eqnarray}
\ln{\left(\tan{\left(\frac{x}{2}\right)}\right)} &=&
\frac{1}{0!}\left[\left.
\frac{\mathrm{d}^0}{\mathrm{d}x^0}
\ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^0
\\&&+\frac{1}{1!}\left[\left.
\frac{\mathrm{d}^1}{\mathrm{d}x^1}
\ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^1
\\&&+\frac{1}{2!}\left[\left.
\frac{\mathrm{d}^2}{\mathrm{d}x^2}
\ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^2
\\&&+\frac{1}{3!}\left[\left.
\frac{\mathrm{d}^3}{\mathrm{d}x^3}
\ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^3
\\&&+\frac{1}{4!}\left[\left.
\frac{\mathrm{d}^4}{\mathrm{d}x^4}
\ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^4
\\&&+\frac{1}{5!}\left[\left.
\frac{\mathrm{d}^5}{\mathrm{d}x^5}
\ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
\right|_{x=\frac{\pi}{2}}\right]\left(x-\frac{\pi}{2}\right)^5
\\&&+\cdots
\\&=&
\frac{1}{1} \left[
\ln{\left(\tan{\left(\frac{\pi}{4}\right)}\right)}
\right]\left(x-\frac{\pi}{2}\right)^0
\\&&+\frac{1}{1}\left[
\frac{1}{\sin{\left(\frac{\pi}{2}\right)}}
\right]\left(x-\frac{\pi}{2}\right)^1
\\&&+\frac{1}{2}\left[
-\frac{\cos{\left(\frac{\pi}{2}\right)}}{\sin^2{\left(\frac{\pi}{2}\right)}}
\right]\left(x-\frac{\pi}{2}\right)^2
\\&&+\frac{1}{6}\left[
\frac{1}{\sin{\left(\frac{\pi}{2}\right)}}
+\frac{2\cos^2{\left(\frac{\pi}{2}\right)}}{\sin^3{\left(\frac{\pi}{2}\right)}}
\right]\left(x-\frac{\pi}{2}\right)^3
\\&&+\frac{1}{24}\left[
-5\frac{\cos{\left(\frac{\pi}{2}\right)}}{\sin^{2}{\left(\frac{\pi}{2}\right)}}
-6\frac{\cos^3{\left(\frac{\pi}{2}\right)}}{\sin^{4}{\left(\frac{\pi}{2}\right)}}
\right]\left(x-\frac{\pi}{2}\right)^4
\\&&+\frac{1}{120}\left[
5\frac{1}{\sin{\left(\frac{\pi}{2}\right)}}
+28\frac{\cos^2{\left(\frac{\pi}{2}\right)}}{\sin^{3}{\left(\frac{\pi}{2}\right)}}
+24\frac{\cos^4{\left(\frac{\pi}{2}\right)}}{\sin^{5}{\left(\frac{\pi}{2}\right)}}
\right]\left(x-\frac{\pi}{2}\right)^5
\\&&+\cdots
\\&=&
\left[0\right]\cdot 1
\\&&+\left[
\frac{1}{1}
\right]\left(x-\frac{\pi}{2}\right)
\\&&+\frac{1}{2}\left[
-\frac{0}{1}
\right]\left(x-\frac{\pi}{2}\right)^2
\\&&+\frac{1}{6}\left[
\frac{1}{1}
+\frac{2\cdot0}{1}
\right]\left(x-\frac{\pi}{2}\right)^3
\\&&+\frac{1}{24}\left[
-5\frac{0}{1}
-6\frac{0}{1}
\right]\left(x-\frac{\pi}{2}\right)^4
\\&&+\frac{1}{120}\left[
5\frac{1}{1}
+28\frac{0}{1}
+24\frac{0}{1}
\right]\left(x-\frac{\pi}{2}\right)^5
\\&&+\cdots
\\&=&
\left(x-\frac{\pi}{2}\right)
+\frac{1}{6}\left(x-\frac{\pi}{2}\right)^3
+\frac{1}{24}\left(x-\frac{\pi}{2}\right)^5+\cdots
\end{eqnarray}$$
\(\lim_{x\rightarrow \frac{\pi}{2}}\ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}\)を求める
$$\begin{eqnarray}
\\\lim_{x\rightarrow \frac{\pi}{2}}\ln{\left(\tan{\left(\frac{x}{2}\right)}\right)}
&=& \lim_{x\rightarrow \frac{\pi}{2}}\left[
\left(x-\frac{\pi}{2}\right)+\frac{1}{6}\left(x-\frac{\pi}{2}\right)^3+\frac{1}{24}\left(x-\frac{\pi}{2}\right)^5+\cdots
\right]
\\&=&0
\end{eqnarray}$$
0 件のコメント:
コメントを投稿