Entire functions with the same zero divisor ($f=e^{h}g,h\mathrm{unique}\mathrm{modulo}2\pi i\mathbb{Z}$)

If two nonzero entire functions have identical zeros and orders, their quotient extends to a nowhere-zero entire function. Its logarithmic derivative has a primitive on the simply connected plane, producing a holomorphic logarithm $h$ and the displayed relation. Any two choices differ by one constant integer multiple of $2\pi i$. On a multiply connected domain the logarithm can fail to exist globally.