  The Fermi function is historically derived from the Dirac equation or the
Schr\"odinger equation. However, we claim that the Fermi function should be
derived from quantum field theory. Then, we obtain the following results: (1)
We give the electromagnetic correction of the beta decay to the fourth order in
$\alpha/v$, where $v$ is the electron velocity. (2) Our result is consistent
with the ordinary non-relativistic Fermi function. (3) We give the iterated
integral representation of the Fermi function for arbitrary order in
$\alpha/v$. (4) This representation is related to the multiple L values, and
they enable us to calculate the amplitude in the algebraic approach.
