What are the consequences if we could express tangent via logarithm in an algebraic system? [closed]

Question

Working on an algebra of divergent integrals I came to the following relation: If $\tau=\int_0^\infty dx$ then

$$\ln (\tau+a)=\int_{0}^\infty \psi'(x+1/2+a)dx$$

and this directly gives the following relation (for finite $z$):

$$\frac1\pi\ln \left(\frac{\tau +\frac{z}{\pi }}{\tau -\frac{z}{\pi }}\right)=\tan z$$

I wonder what interesting consequences there could be of such connection between trigonometric and inverse trigonometric (logarithmic) functions?

P.S. If I did all the manipulations correctly, this gives the following relation:

$$\tan z=\frac2\pi\operatorname{arctanh} \frac{z}{\tau\pi}$$

Answer 1

[This could be a very long comment, if they were allowed]

I think your manipulations may have gone awry somewhere. So your $\tau$ is an infinity that grows like $\Omega(x)$ as $x\to\infty$. One way to approach things would be to replace $\tau$ with $\lim_{x\to\infty}x$ everywhere, argue 'continuity', and pull it out front.

You ought to obtain the exact same results with using an infinitesimal $\epsilon$ for $\frac{1}{\tau}$. But that doesn't seem to work either.

You might be better off starting with simpler identities, based on some written-up formalism that you can point to. There is quite a lot of mathematics on infinities and infinitesimals, done in a variety of formalisms, that works. You just have not given us enough information to help you.