涉及到一个有意思的方程的积分 - Eufisky - The lost book
史济怀复变习题解答

涉及到一个有意思的方程的积分

Eufisky posted @ 2014年7月29日 19:16 in 复分析 with tags 积分计算 , 914 阅读

求证:\begin{align*}&2{\left( {\int_0^\pi  {\frac{x}{{{x^2} + {{\ln }^2}\left( {2\sin x} \right)}}dx} } \right)^7} + 53{\left( {\int_0^\pi  {\frac{x}{{{x^2} + {{\ln }^2}\left( {2\sin x} \right)}}dx} } \right)^5} + {\left( {\int_0^\pi  {\frac{x}{{{x^2} + {{\ln }^2}\left( {2\sin x} \right)}}dx} } \right)^4}\\&+ {\left( {\int_0^\pi  {\frac{x}{{{x^2} + {{\ln }^2}\left( {2\sin x} \right)}}dx} } \right)^3} + 19\int_0^\pi  {\frac{x}{{{x^2} + {{\ln }^2}\left( {2\sin x} \right)}}dx}  = 2014\end{align*}

证明:Here is a proof, using complex analysis, that the integral is equal to $2$. Put

$$f(z) = {1\over\log{(i(1-e^{i2z}))}},$$
the logarithm being the principal branch. As can be deduced from the comments, the original integral is equal to
$$\int_0^\pi \operatorname{Re}{f(x)}\,dx.$$
Next consider, for $R>\epsilon > 0$, the following contour:
 
 
It is straightforward to check that for each fixed $\epsilon$ and $R$, the function $f$ is analytic on an open set containing this contour (just consider where $i(1-e^{i2z})\leq0$; this can only happen when $\operatorname{Im}{z}<0$ or when $\operatorname{Im}{z} = 0$ and $\operatorname{Re}{z}$ is an integer multiple of $\pi$). It then follows from Cauchy's theorem that $f$ integrates to zero over it. First let's see that the integrals over the quarter-circular portions of the contour vanish in the limit $\epsilon \to 0$. I'll look at the quarter circle near $\pi$ (near the bottom right corner of the contour) but the one near zero is similar, if not easier. Writing $i(1-e^{i2z}) = i(1-e^{i2(z-\pi)})$, it is clear that $i(1-e^{i2z}) = O(z-\pi)$ as $z\to\pi$. It follows from this that $f(z) = O(1/\log{|z-\pi|})$ as $z\to0$, and therefore that the integral of $f$ over the bottom right quarter circle is $O(\epsilon/\log{\epsilon})$ as $\epsilon \to0$, hence it vanishes in the limit as claimed.
 
Thus for fixed $R$, we can let $\epsilon \to 0$ to see that $f$ integrates to zero over the rectangle with corners $0,\pi, \pi +iR,$ and $iR$. Now the vertical sides of this rectangle give the contribution
\begin{align*}-\int_0^R f(iy)\,idy + \int_0^R f(iy+\pi)\,idy = 0,\end{align*}
since $f(iy) = f(iy+\pi)$. It follows at once that for each $R$, we can set the contribution from the horizontal sides equal to zero, giving
\begin{align*}\int_0^{\pi} f(x)\,dx - \int_0^\pi f(x+iR)\,dx=0, \qquad R>0. \tag{1}\end{align*}
(Note that the above equation implies that the integral in $x$ of $f(x+iR)$ over the interval $[0,\pi]$ is constant as a function of $R$; another way to evaluate the integral is to prove this directly, which I'll add below in a moment.) Now $f(x+iR) \to 1/\log{i} = 2/\pi$ as $R\to\infty$, uniformly for $x\in[0,\pi]$. Thus
\begin{align*}\lim_{R\to\infty} \int_0^\pi f(x+iR)\,dx = \pi\cdot {2\over \pi} = 2,\end{align*}
and it follows from $(1)$ that the original integral is equal to $2$ as well.
(By the way, the above idea is basically a replication of the technique I used here and here, and which I originally got from Ahlfors' book on complex analysis.)
 

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