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This question is connected to my current research where unexpectedly there arise connections between tronometric/hyperbolic functions and their inverses.

In short, if we introduce some element $\tau$ and a linear operator "standard part" that has the following property:

$$\operatorname{st}(\tau+y)^x=-x\zeta(1-x,1/2+y)$$

for any real or complex numbers $y$ and $x$, we can find the "standard part" of any power and analytic function of $\tau$.

As such, the following interesting relation arises (among others):

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

There are other similarly striking relations.

It is purely technical and can be derived from the mentioned above power rule. I wonder, whether there were other cases when in some algebraic system one could see logarithms be connected to exponents or trigonometric functions to inverse trigonometric in closed form.

This question is connected to my current research where unexpectedly there arise connections between trigonometric/hyperbolic functions and their inverses.

In short, if we introduce some element $\tau$ and a linear operator "standard part" that has the following property:

$$\operatorname{st}(\tau+y)^x=-x\zeta(1-x,1/2+y)$$

for any real or complex numbers $y$ and $x$, we can find the "standard part" of any power and analytic function of $\tau$.

As such, the following interesting relation arises (among others):

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

There are other similarly striking relations.

It is purely technical and can be derived from the mentioned above power rule. I wonder, whether there were other cases when in some algebraic system one could see logarithms be connected to exponents or trigonometric functions to inverse trigonometric in closed form.

This question is connectet to my current research where unexpectedly there arise connections between tronometric/hyperbolic functions and their inverses.

In short, if we introduce some element $\tau$ and a linear operator "standard part" that has the following property:

$$\operatorname{st}(\tau+y)^x=-x\zeta(1-x,1/2+y)$$

for any real or complex numbers $y$ and $x$, we can find the "standard part" of any power and analytic function of $\tau$.

As such, the following interesting relation arises (among others):

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

There are other similarly striking relations.

It purely technical and can be derived from the mentioned above power rule. I wonder, whether there were other cases when in some algebraic system one could see logarithms be connected to exponents or trigonometric functions to inverse trigonometric in closed form.

This question is connected to my current research where unexpectedly there arise connections between tronometric/hyperbolic functions and their inverses.

In short, if we introduce some element $\tau$ and a linear operator "standard part" that has the following property:

$$\operatorname{st}(\tau+y)^x=-x\zeta(1-x,1/2+y)$$

for any real or complex numbers $y$ and $x$, we can find the "standard part" of any power and analytic function of $\tau$.

As such, the following interesting relation arises (among others):

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

There are other similarly striking relations.

It is purely technical and can be derived from the mentioned above power rule. I wonder, whether there were other cases when in some algebraic system one could see logarithms be connected to exponents or trigonometric functions to inverse trigonometric in closed form.