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

$$\ln (\tau+a)=\int_0^\infty \psi'(x+1)dx-\gamma + \psi(a+1/2)$$

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:

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

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:

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

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

$$\ln (\tau+a)=\int_0^\infty \psi'(x+1)dx + \psi(a+1/2)$$

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:

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

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

$$\ln (\tau+a)=\int_0^\infty \psi'(x+1)dx-\gamma + \psi(a+1/2)$$

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:

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