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

= 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).