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In calculus, an **antiderivative**, **primitive function**, **primitive integral** or **indefinite integral**
of a function *f* is a differentiable function *F* whose derivative is equal to the original function *f*. This can be stated symbolically as *F*′ = *f*. The process of solving for antiderivatives is called **antidifferentiation** (or **indefinite integration**) and its opposite operation is called differentiation, which is the process of finding a derivative.

Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

The discrete equivalent of the notion of antiderivative is antidifference.

The function *F*(*x*) = *x*^{3}/3 is an antiderivative of *f*(*x*) = *x*^{2}. As the derivative of a constant is zero, *x*^{2} will have an infinite number of antiderivatives, such as *x*^{3}/3, *x*^{3}/3 + 1, *x*^{3}/3 - 2, etc. Thus, all the antiderivatives of *x*^{2} can be obtained by changing the value of C in *F*(*x*) = *x*^{3}/3 + *C*; where *C* is an arbitrary constant known as the constant of integration. Essentially, the graphs of antiderivatives of a given function are vertical translations of each other; each graph's vertical location depending upon the value of *C*.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Antiderivative

In complex analysis, a branch of mathematics, the **antiderivative**, or **primitive**, of a complex-valued function *g* is a function whose complex derivative is *g*. More precisely, given an open set in the complex plane and a function the antiderivative of is a function that satisfies .

As such, this concept is the complex-variable version of the antiderivative of a real-valued function.

The derivative of a constant function is zero. Therefore, any constant is an antiderivative of the zero function. If 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 (those constants need not be equal).

This observation implies that if a function has an antiderivative, then that antiderivative is unique up to addition of a function which is constant on each connected component of .

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* to *b*, the path integral

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Antiderivative_(complex_analysis)

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