Wikipedia:Reference desk/Archives/Mathematics/2015 July 5

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

July 5

zeta function

Why multiplied Riemann equation Gamma by Zeta equation? — Preceding unsigned comment added by 151.236.160.91 (talk) 19:35, 5 July 2015 (UTC)

Because of the identity

$\int _{0}^{\infty }e^{-nx}x^{s-1}\,dx=\Gamma (s)n^{-s}.$

So, summing both sides for $n=1$ to infinity gives:

$\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,dx=\Gamma (s)\zeta (s)$

where $\operatorname {Re} (s)>1$ and $\zeta (s)=\sum _{n=1}^{\infty }n^{-s}$. Sławomir Biały (talk) 21:23, 5 July 2015 (UTC)

Measuring an angle in a letter N

Hello. From time to time, I do projects which involve constructing letters of the alphabet. Most of the time, I measure and eyeball the sides and angles. But I got to wondering about calculating an exact angle. The one that causes me difficulty is the letter N. I try to get the diagonal to line up with the verticals, and I can do it by trial-and-error. But even knowing trigonometric functions, I can't figure out what to input. I measured the angle in red by hand and got approximately 30.743°, but can it be calculated knowing only the other measurements shown? Thank you.    → Michael J Ⓣ Ⓒ Ⓜ 23:02, 5 July 2015 (UTC)

The angle is $\alpha =\tan ^{-1}((21+{\sqrt {57}})/48)=30.7436831\ldots ^{\circ }$. To find it, let x be the vertical distance between the two diagonal lines. You have $x={\frac {1}{\sin \alpha }}$, and also ${\frac {3}{7-x}}=\tan \alpha$. Solving this gives the result. -- Meni Rosenfeld (talk) 23:40, 5 July 2015 (UTC)

I am astonished that your hand measurement was accurate to one-thousandth of a degree! 109.153.225.51 (talk) 23:51, 5 July 2015 (UTC)

I was about to comment the same thing, but User:109.153.225.51 beat me to it. That is some impressively accurate measuring you are doing there! Good job! :) —SeekingAnswers (reply) 09:22, 6 July 2015 (UTC)

Thank you. But I still have a question. How does one determine what is x (The vertical distance between the diagonals, as you say) knowing only the measurements marked? (I would like to know in general terms, in case I need to construct letters of different dimensions.) Thank you.    → Michael J Ⓣ Ⓒ Ⓜ 11:30, 6 July 2015 (UTC)

x is also found from these two equations (two equations in two unknowns. Can be reduced to a quadratic equation using trigonometric identities). In this particular case $x=(3{\sqrt {57}}-7)/8=1.95619\ldots$. To generalize to other shapes, try to figure out why the equations are correct - I can't really explain without drawing, but it shouldn't be hard. The numbers 1, 3 and 7 in the equations are taken from the measurements; the 1 is the diagonal thickness, the thickness of the vertical bars is irrelevant. -- Meni Rosenfeld (talk) 13:28, 6 July 2015 (UTC)

Aha! Now I follow. Thank you.    → Michael J Ⓣ Ⓒ Ⓜ 08:50, 7 July 2015 (UTC)

The claim: " I measured the angle in red by hand and got approximately 30.743° " is more than impressive - it is unbelievable. How did you do it? Bo Jacoby (talk) 11:13, 7 July 2015 (UTC).

Using a graphics program, I created the diagonal bar parallel to the two side bars, then rotated it incrementally, first 1° at a time, then 0.1°, then 0.01° and finally 0.001° (the smallest unit the program allows) until the edge nearly lined up with the corner of the vertical.    → Michael J Ⓣ Ⓒ Ⓜ 14:17, 7 July 2015 (UTC)

That explains it! Thanks. Bo Jacoby (talk) 20:20, 7 July 2015 (UTC).