2
votes
3answers
67 views

Skewed Trigonometric Function

What would be an expression for a periodic function (period $2\pi$) that essentially behaves just like a negative sine function, but it has the following quirk: It's $0$s lie on the usual places ...
5
votes
2answers
88 views

Hyberbolic and Circular (Trig) Functions: Why no parabolic? [duplicate]

There are circular (trig) functions which determine all the points on a unit circle: and which relate to the area swept out by an angle subtended on the circle. -- These functions can of course be ...
3
votes
2answers
67 views

Prove: $\sin (3\pi/2 - x) = -\cos(x)$

I know the sine of $3\pi/2$ is $-1$. So i plug it in the function making it. $\sin(-1-x) = -\cos x$. However, I don't know where to go from here.
0
votes
1answer
21 views

Graph of a Sine Function Increased on One Side

If we want something looks like this sine wave, what is a function that will satisfy this: Further, how can we make this go both ways, meaning either 1) a sine wave staying the same at the bottom ...
0
votes
0answers
49 views

Does $a x+b=\cos(x)$ have a special-functions solution analogous to the Lambert W function?

The Lambert W function is defined as the solution to the equation $z=w e^w$, in the sense that for all $z\in\mathbb C\setminus(-\infty,-1/e]$ we can find a complex number $W(z)$ which obeys ...
1
vote
2answers
205 views

I've seen “hyperbolic rotation” - from this: generalization to multisection rotation: is this possible?

This question is more in recreational mathematics area By accident I came across the concept of "hyperbolic rotation" where we use a matrix containing $\cosh$ and $\sinh$ instead of the ...
9
votes
2answers
202 views

Conjectural closed form for $\int_0^\infty\sqrt[3]z\ \operatorname{Ei}^2(-z)\,dz$

While trying to answer the question "A closed form for $\displaystyle\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx$", I came up with a conjecture: $$\int_0^\infty\sqrt[3]z\ ...
2
votes
3answers
84 views

Need to find function related to Knoedel numbers that satisfies these conditions

I need to find the continuous function $f(x)$ that satisfies $f(0)=0$ and: $$\frac{f(\sin(\pi/6))^2}{\sin^4(\pi/6)}=135$$ $$\frac{f(\sin(\pi/4))^2}{\sin^4(\pi/4)}=63$$ ...
27
votes
3answers
750 views

$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$

Please help me to solve this integral: $$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx.$$ I managed to calculate an indefinite integral of the left part: $$\int\frac{\cos x}{\sin ...
23
votes
1answer
484 views

$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$

I need help with calculating this integral: $$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$ where ...
5
votes
1answer
121 views

Interesting definite integral involving exp and trig

I'm trying to evaluate the following integrals: $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \cos(\phi) d\phi$$ $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \sin(\phi) d\phi$$ for which I want to find ...
5
votes
1answer
160 views

Integral representation of cosecant function

According to Wolfram website http://functions.wolfram.com/ElementaryFunctions/Csc/introductions/Csc/05/, There exists a "well-known" integral representation for the cosecant function, i.e. ...
21
votes
2answers
1k views

Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $

Is there any way to show that $$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}\left( ...
7
votes
1answer
503 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
0
votes
1answer
75 views

Trigonometric term in digamma function $\psi_{0}(-n)$

Solutions to expressions s.a. $$ S(n)=\sum_{k=1}^{n}\frac{1}{k-r} = \psi_{0}(n-r+1)- \psi_{0}(1-r), $$ involves digamma function. For positive values it has the largest term $O(\log(n))$, but ...
2
votes
1answer
346 views

How does $\int_1 ^x \cos(2\pi/t) dt$ have complex values for real values of $x$?

This question is closely related to one I just asked here. I believe that it is just different enough to warrant another question; please let me know if it does not. In the question mentioned above, ...