Power of a holomorphic function using a chosen logarithm

Given a holomorphic logarithm $L$ of a nonvanishing function $f$ , define its power on that branch by

f(z)^a=\exp(aL(z)),\qquad a\in\mathbb C.

It is holomorphic jointly in $z$ and $a$ . Changing the logarithm by $2\pi i k$ multiplies the result by $e^{2\pi i k a}$ , so a noninteger exponent requires a branch choice.

Derivatives and real values

For fixed $a$ , $(f^a)'=a f^a f'/f$ , and $\partial_a f^a=L f^a$ . If $f$ is positive on a real interval and $L$ agrees with the real logarithm there, this definition agrees with the real positive-base power. Derivative estimates require a bound on the chosen branch in a complex neighborhood.