In complex analysis
, a branch of mathematics
, the antiderivative
, or primitive
, of a complex
-valued function g
is a function whose complex derivative
. More precisely, given an open set
<math>U</math> in the complex plane and a function <math>g:Uto mathbb C,</math> the antiderivative of <math>g</math> is a function <math>f:Uto mathbb C</math> that satisfies <math>frac=g</math>.
As such, this concept is the complex-variable version of the antiderivative
of a real
The derivative of a constant function is zero. Therefore, any constant is an antiderivative of the zero function. If <math>U</math> is a connected set
, then the constants are the only antiderivatives of the zero function. Otherwise, a function is an antiderivative of the zero function if and only if it is constant on each connected component
of <math>U</math> (those constants need not be equal).
This observation implies that if a function <math>g:Uto mathbb C</math> has an antiderivative, then that antiderivative is unique up to
addition of a function which is constant on each connected component of <math>U</math>.
One can characterize the existence of antiderivatives via path integrals in the complex plane, much like the case of functions of a real variable. Perhaps not surprisingly, g
has an antiderivative f
if and only if, for every γ
path from a
... Read More