Appendix C --- What is the electron spin?

Appendix C

Zeta-exponential function

As we know the exponential function is:

The Riemann Zeta function is:

Combine the exponential function and zeta function, and then we can have a new function, which we temporally called the zeta-exponential function as follows:

Then we will get:

Thus:

Thus, we have:

The above equation is the Fourier serial of the complex zeta-exponential function.

As we know:

Thus:

As we know for the exponential function, we have:

Similar to the exponential function, the complex zeta-exponential function has the following relationship:

for m =1 and , thus we have