Sinh(sinh(i+sinh(i+sinh(i+sinh(i+...)))))

Today I was trying to find the value of 2^i, not exact value though but something in the form of a formula. After 3 hours I ended up with a generalized formula for n^i. I am attaching an image and a pdf file of my work.

I run a Facebook Group called not just tiny-c Programming Group. A member recently posted the above problem. This is actually a very old problem but we were encouraged to give it a shot without looking it up. The original problem was this: graph the above equation. For "extra credit" we were encouraged to solve it for rational numbers. An obvious solution is x=y. Also, 2^4=4^2. It turns out there are infinitely many rational numbers that solve this equation. I wrote a computer program in C which approximately solved our equation. I populated a table with the x,y values it generated into desmos and got a pretty good graph. Actually, you can just put the equation itself into desmos and get the exact graph. But the rationals was too tough for me. I eventually did the research and got the answer. Very, very interesting. I watched blackpenredpen's videos on this. I doubt this rational problem would have enough general interest to make a video on this but ... you never know.

I happened upon this in MIT OpenCourseWare: https://www.youtube.com/watch?v=PNKj529yY5c&feature=youtu.be&t=963 This transform allows converting an integral of the form (tan(x))^n to a polynomial problem. For example: Let: f( tan x ) = ( tan x )^4 and y = tan x Then: f( y ) = ( y )^4 Substituting into the transform: Integral [tan^4 (x) dx] = Integral [ y^4 / (1 + y^2) dy ] Using long division: y^4 / (1 + y^2) = y^2 - 1 + 1 / (1 + y^2) Substituting into the right-hand-side of the transformed problem above and performing the integration: Integral [ y^2 - 1 + 1 / (1 + y^2) dy ] = 1/3 * y^3 - y + Arctan(y) Then returning to the x-world: Integral [tan^4 (x) dx] = 1/3 * tan^3(x) - tan(x) + x + C He said in the video: "there's a whole family of things like that," but I don't know where. My Google searches didn't point me to a list where this transform is included. Are other members of this family available online somewhere? Maybe this family could be printed on T-shirts?