Integral representation of the Euler-Mascheroni constant involving $\pi$

So here is my proof:

Two important things that i wont proof here are

$\lfloor x\rfloor = \frac {\arctan(\cot(x\pi))}\pi - \frac 12 + x$ and

$\sum_{n=1}^x \frac 1n = \frac {\lfloor x\rfloor}x + \int_1^x \frac {\lfloor t\rfloor}{t^2} \,dt$

But those are well known facts.

From this it follows that

$\sum_{n=1}^x \frac 1n = \frac {\arctan(\cot(x\pi))}{\pi x} - \frac 1{2x} + 1 + \int_1^x \frac {\arctan(\cot(t\pi))}{\pi t^2} - \frac 1{2t^2} + \frac 1t\,dt$

(simply substitute the first into the second)

$\sum_{n=1}^x \frac 1n = \frac {\arctan(\cot(x\pi))}{\pi x} - \frac 1{2x} + 1 + \frac 1{2x} - \frac 12 + \ln(x) + \int_1^x \frac {\arctan(\cot(t\pi))}{\pi t^2} \,dt$

$\sum_{n=1}^x \frac 1n - \ln(x) = \frac 12 + \int_1^x \frac {\arctan(\cot(t\pi))}{\pi t^2} \,dt + \frac {\arctan(cot(x\pi))}{\pi x}$

Now take the limit as x goes to $\infty$

$\gamma = \frac 12 + \int_1^\infty \frac {\arctan(\cot(t\pi))}{\pi t^2} \,dt$

substituting in $u = \frac 1{\pi t}$ gives

$\gamma = \frac 12 + \int_0^{\frac 1{\pi}} \arctan(\cot(\frac 1u)) \,du$

Tags:

Euler Mascheroni Constant

Related

Compute $\int_{\mathbb R}\frac{dx}{1+2x^2+x^4}$ $ \int_0^\frac{\pi}{2}\ln^n\left(\tan(x)\right)\:dx$ It is impossible for $(x-1)^2+x^2+(x+1)^2$ to be a perfect square Express $\sum_{j=1}^{n}\sum_{i=1}^{n} \frac{1}{i(i+j)}$ in terms of harmonic numbers Every interval $I \subset \mathbb{R}$ is connected. [Proof clarification] Is this a scalar product of three vectors? Has $n^{n+1}+(n+1)^{n+2}$ other obvious factors than that I found? Can I get a sequence of bounded functions converging pointwise to $f(x)=1/x$ for $x$ non zero and $0$ for $x$ zero? countable family of open sets Unclear ideas in the proof of the Archimedean Principle Given $a,b,c$ positive numbers, is there a function $h$ such that $h(ax+b) = c \cdot h(x)$ for all $x>0$? How do we know $\sin$ and $\cos$ are the only solutions to $y'' = -y$?

Recent Posts