로그 탄젠트 적분(log tangent integral)

<H5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em">이 항목의 스프링노트 원문주소</H5> <UL> <LI> <A class=wiki title="로그 탄젠트 적분(log tangent integral)" href="http://pythagoras0.springnote.com/pages/3989775">로그 탄젠트 적분(log tangent integral)</A></LI></UL>     <H5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em">개요</H5> <UL> <LI> <A class=wiki title="로그 사인 적분 (log sine integrals)" href="http://pythagoras0.springnote.com/pages/5896529">로그 사인 적분 (log sine integrals)</A>과 밀접하게 관련되어 있음</LI> <LI> 다음과 같은 정적분값의 계산 <IMG class=equation alt="\int_{\pi/4}^{\pi/2} \ln \tan x\, dx=" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cint_%257B%255Cpi%252F4%257D%255E%257B%255Cpi%252F2%257D%2520%255Cln%2520%255Ctan%2520x%255C%252C%2520dx%253DG">, <IMG class=equation alt=G src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3DG">는 <A class=wiki title="카탈란 상수" href="http://pythagoras0.springnote.com/pages/4206061">카탈란 상수</A> <IMG class=equation alt="\int_{0}^{\pi/4} \ln \tan x\,dx=-G" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cint_%257B0%257D%255E%257B%255Cpi%252F4%257D%2520%255Cln%2520%255Ctan%2520x%255C%252Cdx%253D-G"> <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt="\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cint_%257B%255Cpi%252F4%257D%255E%257B%255Cpi%252F2%257D%2520%255Cln%2520%255Cln%2520%255Ctan%2520x%255C%252C%2520dx%253D%255Cfrac%257B%255Cpi%257D%257B2%257D%255Cln%257B%255Cfrac%257B%255CGamma%2528%255Cfrac%257B3%257D%257B4%257D%2529%257D%257B%255CGamma%2528%255Cfrac%257B1%257D%257B4%257D%2529%257D%255Csqrt%257B2%255Cpi%257D"> <IMG class=equation alt="\int_{0}^{\pi} \ln^2 \tan \frac{x}{4}\,dx=\frac{\pi^3}{4}" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cint_%257B0%257D%255E%257B%255Cpi%257D%2520%255Cln%255E2%2520%255Ctan%2520%255Cfrac%257Bx%257D%257B4%257D%255C%252Cdx%253D%255Cfrac%257B%255Cpi%255E3%257D%257B4%257D"> <IMG class=equation alt="\int_0^{\infty}\frac{\ln^2 x}{1+x^2} dx =\int_{0}^{\pi/2}\ln^2 \tan x\,dx = \frac{ \pi^3}{8}" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cint_0%255E%257B%255Cinfty%257D%255Cfrac%257B%255Cln%255E2%2520x%257D%257B1%252Bx%255E2%257D%2520dx%2520%253D%255Cint_%257B0%257D%255E%257B%255Cpi%252F2%257D%255Cln%255E2%2520%255Ctan%2520x%255C%252Cdx%2520%253D%2520%255Cfrac%257B%2520%255Cpi%255E3%257D%257B8%257D"></LI></UL>     <H5 style="LINE-HEIGHT: 2em; MARGIN: 0px">증명</H5> [Vardi1988] 참조  (보조정리) <IMG class=equation alt="\Gamma(s)\beta(s)=\int_{\pi/4}^{\pi/2} \ln^{s-1}\tan x\, dx" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255CGamma%28s%29%255Cbeta%28s%29%253D%255Cint_%257B%255Cpi%252F4%257D%255E%257B%255Cpi%252F2%257D%2520%2520%255Cln%255E%257Bs-1%257D%255Ctan%2520x%255C%252C%2520dx"> 여기서 <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=\Gamma(s) src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255CGamma%2528s%2529">는 <A style="LINE-HEIGHT: 2em; COLOR: rgb(0,43,184) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=wiki title=감마함수 href="http://pythagoras0.springnote.com/pages/3197800">감마함수</A>,<IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=\beta(s) src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cbeta%2528s%2529">는 <A style="LINE-HEIGHT: 2em; COLOR: rgb(0,43,184) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=wiki title="디리클레 베타함수" href="http://pythagoras0.springnote.com/pages/4206067">디리클레 베타함수</A>.   (증명) <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s} src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3DF%2528s%2529%253D%255Csum_%257Bn%253D1%257D%255E%257B%255Cinfty%257D%255Cfrac%257Bf%2528n%2529%257D%257Bn%255Es%257D"> 라 하자. <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=\Gamma(s)F(s)=\int_0^{\infty}(\sum_{n=1}^{\infty}f(n)e^{-nt})t^{s-1}\,dt src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255CGamma%2528s%2529F%2528s%2529%253D%255Cint_0%255E%257B%255Cinfty%257D%2528%255Csum_%257Bn%253D1%257D%255E%257B%255Cinfty%257Df%2528n%2529e%255E%257B-nt%257D%2529t%255E%257Bs-1%257D%255C%252Cdt"> <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=z=e^{-t} src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3Dz%253De%255E%257B-t%257D"> 로 치환하면, <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=\Gamma(s)F(s)=\int_0^{1}(\sum_{n=1}^{\infty}f(n)z^n)(\log\frac{1}{z})^{s-1}\,\frac{dz}{z} src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255CGamma%2528s%2529F%2528s%2529%253D%255Cint_0%255E%257B1%257D%2528%255Csum_%257Bn%253D1%257D%255E%257B%255Cinfty%257Df%2528n%2529z%255En%2529%2528%255Clog%255Cfrac%257B1%257D%257Bz%257D%2529%255E%257Bs-1%257D%255C%252C%255Cfrac%257Bdz%257D%257Bz%257D">   만약 <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=f(n+q)=f(n) src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3Df%2528n%252Bq%2529%253Df%2528n%2529"> 을 만족하면 (가령 디리클레 캐릭터의 경우) <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=p(z)=\sum_{n=1}^{q-1}f(n)z^n src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3Dp%2528z%2529%253D%255Csum_%257Bn%253D1%257D%255E%257Bq-1%257Df%2528n%2529z%255En">라면,  <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=\sum_{n=1}^{\infty}f(n)z^n=\frac{p(z)}{1-z^q} src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Csum_%257Bn%253D1%257D%255E%257B%255Cinfty%257Df%2528n%2529z%255En%253D%255Cfrac%257Bp%2528z%2529%257D%257B1-z%255Eq%257D"> 로 쓸 수 있다.   이를 이용하면,  <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=\Gamma(s)F(s)=\int_0^{1}\frac{p(z)(\log\frac{1}{z})^{s-1}}{1-z^q}\,\frac{dz}{z} src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255CGamma%2528s%2529F%2528s%2529%253D%255Cint_0%255E%257B1%257D%255Cfrac%257Bp%2528z%2529%2528%255Clog%255Cfrac%257B1%257D%257Bz%257D%2529%255E%257Bs-1%257D%257D%257B1-z%255Eq%257D%255C%252C%255Cfrac%257Bdz%257D%257Bz%257D"> 를 얻는다. <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=f src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3Df">가 <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=f(3)=-1 src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3Df%25283%2529%253D-1">인 주기가 4인 디리클레 캐릭터라면, <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=q=4 src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3Dq%253D4">, <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=p(z)=z-z^3 src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3Dp%2528z%2529%253Dz-z%255E3"> 따라서 <IMG class=equation alt="\Gamma(s)\beta(s)=\int_0^{1}\frac{(\log\frac{1}{z})^{s-1}}{1+z^2} \,dz=\int_1^{\infty}\frac{(\log u)^{s-1}}{1+u^2} \,du=\int_{\pi/4}^{\pi/2} \ln^{s-1}\tan x\, dx" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255CGamma%28s%29%255Cbeta%28s%29%253D%255Cint_0%255E%257B1%257D%255Cfrac%257B%28%255Clog%255Cfrac%257B1%257D%257Bz%257D%29%255E%257Bs-1%257D%257D%257B1%252Bz%255E2%257D%2520%2520%255C%252Cdz%253D%255Cint_1%255E%257B%255Cinfty%257D%255Cfrac%257B%28%255Clog%2520u%29%255E%257Bs-1%257D%257D%257B1%252Bu%255E2%257D%2520%2520%255C%252Cdu%253D%255Cint_%257B%255Cpi%252F4%257D%255E%257B%255Cpi%252F2%257D%2520%2520%255Cln%255E%257Bs-1%257D%255Ctan%2520x%255C%252C%2520dx"> ■     (따름정리1) <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt="\int_{\pi/4}^{\pi/2} \ln \tan x\, dx=" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cint_%257B%255Cpi%252F4%257D%255E%257B%255Cpi%252F2%257D%2520%255Cln%2520%255Ctan%2520x%255C%252C%2520dx%253DG">, G는 <A class=wiki title="카탈란 상수" href="http://pythagoras0.springnote.com/pages/4206061">카탈란 상수</A>. (증명) 위에서 얻은 보조정리에 <IMG class=equation alt=s=2 src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3Ds%253D2">를 적용하면,  <IMG class=equation alt="\int_{\pi/4}^{\pi/2} \ln^{2-1}\tan x\, dx=" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cint_%257B%255Cpi%252F4%257D%255E%257B%255Cpi%252F2%257D%2520%255Cln%255E%257B2-1%257D%255Ctan%2520x%255C%252C%2520dx%253D%255CGamma%282%29%255Cbeta%282%29%253DG"> ■     (따름정리2) <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt="\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cint_%257B%255Cpi%252F4%257D%255E%257B%255Cpi%252F2%257D%2520%255Cln%2520%255Cln%2520%255Ctan%2520x%255C%252C%2520dx%253D%255Cfrac%257B%255Cpi%257D%257B2%257D%255Cln%257B%255Cfrac%257B%255CGamma%2528%255Cfrac%257B3%257D%257B4%257D%2529%257D%257B%255CGamma%2528%255Cfrac%257B1%257D%257B4%257D%2529%257D%255Csqrt%257B2%255Cpi%257D">   (증명) <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt="\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cint_%257B%255Cpi%252F4%257D%255E%257B%255Cpi%252F2%257D%2520%255Cln%2520%255Cln%2520%255Ctan%2520x%255C%252C%2520dx%253D%255Cfrac%257Bd%257D%257Bds%257D%28%255CGamma%28s%29%255Cbeta%28s%29%29%257C_%257Bs%253D1%257D">임을 보이자. <IMG class=equation alt="\frac{d}{ds}(\Gamma(s)\beta(s))=\frac{d}{ds}\int_1^{\infty}\frac{(\log u)^{s-1}}{1+u^2} \,du=\int_1^{\infty}\frac{(\log u)^{s-1}}{1+u^2}\log \log u \,du" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cfrac%257Bd%257D%257Bds%257D%28%255CGamma%28s%29%255Cbeta%28s%29%29%253D%255Cfrac%257Bd%257D%257Bds%257D%255Cint_1%255E%257B%255Cinfty%257D%255Cfrac%257B%28%255Clog%2520u%29%255E%257Bs-1%257D%257D%257B1%252Bu%255E2%257D%2520%255C%252Cdu%253D%255Cint_1%255E%257B%255Cinfty%257D%255Cfrac%257B%28%255Clog%2520u%29%255E%257Bs-1%257D%257D%257B1%252Bu%255E2%257D%255Clog%2520%255Clog%2520u%2520%255C%252Cdu" width=466 height=39> <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt=s=1 src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3Ds%253D1"> 일때, <IMG class=equation alt="\Gamma'(1)\beta(1)+\Gamma(1)\beta'(1)=\int_1^{\infty}\log \log u \,\frac{du}{1+u^2}=\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255CGamma'(1)%5Cbeta(1)%2B%5CGamma(1)%5Cbeta'(1)%3D%5Cint_1%5E%7B%5Cinfty%7D%5Clog%20%5Clog%20u%20%5C%2C%5Cfrac%7Bdu%7D%7B1%2Bu%5E2%7D%3D%5Cint_%7B%5Cpi%2F4%7D%5E%7B%5Cpi%2F2%7D%20%5Cln%20%5Cln%20%5Ctan%20x%5C%2C%20dx"> 이제 <A style="LINE-HEIGHT: 2em; COLOR: rgb(0,43,184) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=wiki title="Digamma 함수" href="http://pythagoras0.springnote.com/pages/3767493">Digamma 함수</A>와 <A style="LINE-HEIGHT: 2em; COLOR: rgb(0,43,184) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=wiki title="디리클레 베타함수" href="http://pythagoras0.springnote.com/pages/4206067">디리클레 베타함수</A>에서 얻은 결과를 사용하자.    <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt="\psi(x) =\frac{d}{dx} \ln{\Gamma(x)}= \frac{\Gamma'(x)}{\Gamma(x)}" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cpsi%2528x%2529%2520%253D%255Cfrac%257Bd%257D%257Bdx%257D%2520%255Cln%257B%255CGamma%2528x%2529%257D%253D%2520%255Cfrac%257B%255CGamma%2527%2528x%2529%257D%257B%255CGamma%2528x%2529%257D">, <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt="\psi(1) = -\gamma\,\!" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cpsi%25281%2529%2520%253D%2520-%255Cgamma%255C%252C%255C%2521">. 따라서 <IMG class=equation alt=\Gamma(1)=-\gamma src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255CGamma%281%29%253D-%255Cgamma">. <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt="\beta'(1)=\frac{\pi}{4}\gamma+\frac{\pi}{2}\ln(\frac{\Gamma(3/4)}{\Gamma(1/4)}\sqrt{2\pi})" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cbeta%2527%25281%2529%253D%255Cfrac%257B%255Cpi%257D%257B4%257D%255Cgamma%252B%255Cfrac%257B%255Cpi%257D%257B2%257D%255Cln%2528%255Cfrac%257B%255CGamma%25283%252F4%2529%257D%257B%255CGamma%25281%252F4%2529%257D%255Csqrt%257B2%255Cpi%257D%2529">.   그러므로 <IMG class=equation alt="\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cint_%257B%255Cpi%252F4%257D%255E%257B%255Cpi%252F2%257D%2520%255Cln%2520%255Cln%2520%255Ctan%2520x%255C%252C%2520dx%253D%255CGamma'(1)%5Cbeta(1)%2B%5CGamma(1)%5Cbeta'(1)%3D%20-%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cgamma%2B%5Cbeta'(1)%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cln%7B%5Cfrac%7B%5CGamma(%5Cfrac%7B3%7D%7B4%7D)%7D%7B%5CGamma(%5Cfrac%7B1%7D%7B4%7D)%7D%5Csqrt%7B2%5Cpi%7D"> 임이 증명된다. ■     <H5 style="LINE-HEIGHT: 2em; MARGIN: 0px">메모</H5> <IMG class=equation alt=\int_{0}^{\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}\,dx=\pi\ln2 src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cint_%257B0%257D%255E%257B%255Cinfty%257D%255Cfrac%257B%255Cln%28x%255E%257B2%257D%252B1%29%257D%257Bx%255E%257B2%257D%252B1%257D%255C%252Cdx%253D%255Cpi%255Cln2"> <UL> <LI> <A class=wiki title="로그 사인 적분 (log sine integrals)" href="http://pythagoras0.springnote.com/pages/5896529">로그 사인 적분 (log sine integrals)</A></LI> <LI> <A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" href="http://cjackal.tistory.com/109">http://cjackal.tistory.com/10</A><A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" href="http://cjackal.tistory.com/109">9</A></LI> <LI> <A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" href="http://www.artofproblemsolving.com/Forum/viewtopic.php?t=340081">http://www.artofproblemsolving.com/Forum/viewtopic.php?t=340081</A></LI></UL>   <IMG style="BORDER-BOTTOM: 0px; BORDER-LEFT: 0px; LINE-HEIGHT: 2em; BORDER-TOP: 0px; BORDER-RIGHT: 0px" class=equation alt="\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt=L_{-7}(2)=1.15192547054449\cdots" src="https://img1.daumcdn.net/relay/cafe/original/?fname=http%3A%2F%2Feq.springnote.com%2Ftex_image%3Fsource%3D%255Cfrac%257B24%257D%257B7%255Csqrt%257B7%257D%257D%255Cint_%257B%255Cpi%252F3%257D%255E%257B%255Cpi%252F2%257D%255Cln%257C%255Cfrac%257B%255Ctan%2520t%252B%255Csqrt%257B7%257D%257D%257B%255Ctan%2520t-%255Csqrt%257B7%257D%257D%257C%255C%252Cdt%253DL_%257B-7%257D%282%29%253D1.15192547054449%255Ccdots">     <A style="LINE-HEIGHT: 2em; COLOR: rgb(0,43,184) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=wiki title="란덴변환(Landen's transformation)" href="http://pythagoras0.springnote.com/pages/2998854"></A> <H5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em">재미있는 사실</H5>     <H5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em">역사</H5> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em"><A style="LINE-HEIGHT: 2em; COLOR: rgb(0,43,184) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=wiki title=수학사연표 href="http://pythagoras0.springnote.com/pages/3304643">수학사연표</A></LI></UL>     <H5 style="LINE-HEIGHT: 2em; MARGIN: 0px">메모</H5> <UL> <LI> <A href="http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.0356v1.pdf">http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.0356v1.pdf</A></LI></UL> <UL> <LI> <A href="http://www.wolframalpha.com/input/?i=integrate_0%5E%28pi%29+x+cos+x+%2F%281%2Bsin%5E2+x%29">http://www.wolframalpha.com/input/?i=integrate_0^(pi)+x+cos+x+%2F(1%2Bsin^2+x)</A></LI> <LI> <A href="http://www.wolframalpha.com/input/?i=log%5E2+%281%2Bsqrt%282%29%29+-pi%5E2%2F4">http://www.wolframalpha.com/input/?i=log^2+(1%2Bsqrt(2))+-pi^2%2F4</A></LI> <LI> <A href="http://www.wolframalpha.com/input/?i=integrate+%28tan+x%2B1%29/sqrt%28tan%5E2+x+%2B1%29dx">http://www.wolframalpha.com/input/?i=integrate+(tan+x%2B1)/sqrt(tan^2+x+%2B1)dx</A></LI> <LI> <A href="http://www.wolframalpha.com/input/?i=integrate+%28tan+x-1%29/sqrt%28tan%5E2+x+%2B1%29dx">http://www.wolframalpha.com/input/?i=integrate+(tan+x-1)/sqrt(tan^2+x+%2B1)dx</A></LI></UL>     <H5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em">관련된 다른 주제들</H5> <UL> <LI> <A class=wiki title="등차수열의 소수분포에 관한 디리클레 정리" href="http://pythagoras0.springnote.com/pages/3154294">등차수열의 소수분포에 관한 디리클레 정리</A></LI> <LI> <A class=wiki title="디리클레 급수" href="http://pythagoras0.springnote.com/pages/4181559">디리클레 급수</A></LI> <LI> <A class=wiki title="Hurwitz 제타함수" href="http://pythagoras0.springnote.com/pages/4075075">Hurwitz 제타함수</A></LI> <LI> <A class=wiki title=감마함수 href="http://pythagoras0.springnote.com/pages/3197800">감마함수</A></LI> <LI> <A class=wiki title="다이로그 함수(dilogarithm )" href="http://pythagoras0.springnote.com/pages/3321277">다이로그 함수(dilogarithm )</A></LI> <LI> <A style="LINE-HEIGHT: 2em; COLOR: rgb(0,43,184) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=wiki title="란덴변환(Landen's transformation)" href="http://pythagoras0.springnote.com/pages/2998854">란덴변환(Landen's transformation)</A></LI> <LI> <A class=wiki title="르장드르 카이 함수" href="http://pythagoras0.springnote.com/pages/4372045">르장드르 카이 함수</A></LI> <LI> <A class=wiki title="모듈라 군, j-invariant and the singular moduli" href="http://pythagoras0.springnote.com/pages/1950432">모듈라 군, j-invariant and the singular moduli</A></LI> <LI> <A style="LINE-HEIGHT: 2em; COLOR: rgb(0,43,184) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=wiki title="카탈란 상수" href="http://pythagoras0.springnote.com/pages/4206061">카탈란 상수</A></LI></UL>     <H5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em">수학용어번역</H5> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em"> <A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=external title=http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= href="http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr=">대한수학회 수학 학술 용어집</A> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em">http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=</LI></UL></LI> <LI style="LINE-HEIGHT: 2em"><A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=external title=http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid={D6048897-56F9-43D7-8BB6-50B362D1243A}&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 href="http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4">대한수학회 수학용어한글화 게시판</A></LI></UL>     <H5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em">사전 형태의 자료</H5> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em">http://ko.wikipedia.org/wiki/</LI> <LI style="LINE-HEIGHT: 2em">http://en.wikipedia.org/wiki/</LI> <LI style="LINE-HEIGHT: 2em">http://www.wolframalpha.com/input/?i=</LI> <LI style="LINE-HEIGHT: 2em"><A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=external title=http://dlmf.nist.gov href="http://dlmf.nist.gov/">NIST Digital Library of Mathematical Functions</A></LI></UL>     <H5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em">관련논문</H5> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em"> <A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=external title=http://dx.doi.org/10.1007/s11040-010-9074-y href="http://dx.doi.org/10.1007/s11040-010-9074-y">Alternative Eval‎uation of a ln tan Integral Arising in Quantum Field Theory</A> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em"> Mark W. Coffey, Mathematical Physics, Analysis and Geometry, Volume 13, Number 2, 2010</LI></UL></LI> <LI style="LINE-HEIGHT: 2em">   [BBBZ2010]<A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=external title=http://arxiv.org/abs/1005.0414 href="http://arxiv.org/abs/1005.0414">Experimental Mathematics and Mathematical Physics</A> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em">DH Bailey, JM Borwein, D Broadhurst, W Zudilin, 2010</LI></UL></LI> <LI style="LINE-HEIGHT: 2em"> <A class=external title=http://www.springerlink.com/content/p2k0106727416271/?p=03915f5244d74523b6d36406299c80d5&pi=6 href="http://www.springerlink.com/content/p2k0106727416271/?p=03915f5244d74523b6d36406299c80d5&pi=6">A class of logarithmic integrals</A> <UL> <LI style="LINE-HEIGHT: 2em">Luis A. Medina and Victor H. Moll, The Ramanujan Journal, Volume 20, Number 1 / 2009년 10월</LI></UL></LI> <LI> <A class=external title=http://link.aip.org/link/?JMAPAQ/49/093508/1 href="http://link.aip.org/link/?JMAPAQ/49/093508/1">Eval‎uation of a ln tan integral arising in quantum field theory</A> <UL> <LI>Mark W. Coffey, J. Math. Phys. 49, 093508 (2008)</LI></UL></LI> <LI style="LINE-HEIGHT: 2em"> [Borwein and Boradhurst 1998]<A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=external title=http://arxiv.org/abs/hep-th/9811173 href="http://arxiv.org/abs/hep-th/9811173">Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links</A> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em">J.M. Borwein, D.J. Broadhurst, 1998</LI></UL></LI> <LI style="LINE-HEIGHT: 2em"> <A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=external title=http://doi.acm.org/10.1145/258726.258736 href="http://doi.acm.org/10.1145/258726.258736">A class of logarithmic integrals</A> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em">Victor Adamchik, 1997</LI></UL></LI> <LI style="LINE-HEIGHT: 2em"> [Vardi1988]<A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=external title=http://www.jstor.org/stable/2323562 href="http://www.jstor.org/stable/2323562">Integrals, an Introduction to Analytic Number Theory</A> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em">Ilan Vardi, The American Mathematical Monthly, Vol. 95, No. 4 (Apr., 1988), pp. 308-315</LI></UL></LI></UL>     <H5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em">관련도서 및 추천도서</H5> <UL> <LI> <A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=external title=http://www.amazon.com/Irresistible-Integrals-Symbolics-Experiments-Eval‎uation/dp/0521796369 href="http://www.amazon.com/Irresistible-Integrals-Symbolics-Experiments-Eval‎uation/dp/0521796369">Irresistible Integrals: Symbolics, Analysis and Experiments in the Eval‎uation of Integrals</A> <UL> <LI style="LINE-HEIGHT: 2em">George Boros and Victor Moll</LI></UL></LI></UL> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em"> 도서내검색 <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em">http://books.google.com/books?q=</LI> <LI style="LINE-HEIGHT: 2em">http://book.daum.net/search/contentSearch.do?query=</LI></UL></LI> <LI style="LINE-HEIGHT: 2em"> 도서검색 <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em">http://books.google.com/books?q=</LI> <LI style="LINE-HEIGHT: 2em">http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=</LI> <LI style="LINE-HEIGHT: 2em">http://book.daum.net/search/mainSearch.do?query=</LI></UL></LI></UL>     <H5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em">관련기사</H5> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em"> 네이버 뉴스 검색 (키워드 수정) <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em">http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=</LI> <LI style="LINE-HEIGHT: 2em">http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=</LI> <LI style="LINE-HEIGHT: 2em">http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=</LI></UL></LI></UL>     <H5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em">블로그</H5> <UL style="PADDING-BOTTOM: 0px; LINE-HEIGHT: 2em; MARGIN: 0px; PADDING-LEFT: 24px; PADDING-RIGHT: 0px; PADDING-TOP: 0px"> <LI style="LINE-HEIGHT: 2em">구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=</LI> <LI style="LINE-HEIGHT: 2em"><A style="LINE-HEIGHT: 2em; COLOR: rgb(86,137,66) !important; CURSOR: pointer !important; TEXT-DECORATION: underline" class=external title=http://navercast.naver.com/science/list href="http://navercast.naver.com/science/list">네이버 오늘의과학</A></LI></UL>