Antiderivative (complex analysis)
Encyclopedia
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.
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
.
However, this formal similarity notwithstanding, possessing a complex-antiderivative is a much more restrictive condition than its real counterpart. While it is possible for a discontinuous real function to have an anti-derivative, anti-derivatives can fail to exist even for holomorphic functions
of a complex variable. For example, consider the reciprocal function,
which is holomorphic on the punctured plane C\{0}. A direct calculation shows that the integral of g along any circle enlosing the origin is non-zero. So g fails the condition cited above.
In fact, holomorphy is characterized by having an antiderivative locally, that is, g is holomorphic if for every z in its domain, there is some neighborhood U of z such that g has an antiderivative on U. Furthermore, holomorphy is a necessary condition for a function to have an antiderivative, since the derivative of any holomorphic function is holomorphic.
Various versions of Cauchy integral theorem, an underpinning result of Cauchy function theory, which makes heavy use of path integrals, gives sufficient conditions under which, for a holomorphic g, ∫γ g(ζ) d ζ does vanish for any closed path γ (which may be, for instance, that the domain of g be simply connected or star-convex).
is an antiderivative of 
on 
, then it has the path integral property given above. Given any piecewise C1
path
, one can express the path integral
of
over 
as
By the chain rule
and the fundamental theorem of calculus
one then has
Therefore the integral of
over 
does not depend on the actual path 
, but only on its endpoints, which is what we wanted to show.
over any path depends only on the endpoints, then g has an antiderivative. We will do so by finding an anti-derivative explicitly.
Without loss of generality, we can assume that the domain
of 
is connected, as otherwise one can prove the existence of an antiderivative on each connected component. With this assumption, fix a point 
in 
and for any 
in 
define the function
where
is any path joining 
to 
. Such a path exists since 
is assumed to be an open connected set. The function 
is well-defined because the integral depends only on the endpoints of 
.
That this
is an antiderivative of 
can be argued in the same way as the real case. We have, for a given z in U,
where [z, w] denotes the line segment between z and w. By continuity of g, the final expression goes to zero as w approaches z. In other words, f' = g.
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the antiderivative, or primitive, of a complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
-valued function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
g is a function whose complex derivative is g. More precisely, given an open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
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
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
of a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
-valued function.
Uniqueness
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 toUp to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
addition of a function which is constant on each connected component of
.Existence
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 ∫γ g(ζ) d ζ = f(b) - f(a). Equivalently, ∫γ g(ζ) d ζ = 0 for any closed path γ.However, this formal similarity notwithstanding, possessing a complex-antiderivative is a much more restrictive condition than its real counterpart. While it is possible for a discontinuous real function to have an anti-derivative, anti-derivatives can fail to exist even for holomorphic functions
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
of a complex variable. For example, consider the reciprocal function,
which is holomorphic on the punctured plane C\{0}. A direct calculation shows that the integral of g along any circle enlosing the origin is non-zero. So g fails the condition cited above.In fact, holomorphy is characterized by having an antiderivative locally, that is, g is holomorphic if for every z in its domain, there is some neighborhood U of z such that g has an antiderivative on U. Furthermore, holomorphy is a necessary condition for a function to have an antiderivative, since the derivative of any holomorphic function is holomorphic.
Various versions of Cauchy integral theorem, an underpinning result of Cauchy function theory, which makes heavy use of path integrals, gives sufficient conditions under which, for a holomorphic g, ∫γ g(ζ) d ζ does vanish for any closed path γ (which may be, for instance, that the domain of g be simply connected or star-convex).
Necessity
First we show that if is an antiderivative of on , then it has the path integral property given above. Given any piecewise C1Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
path
Path (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
, one can express the path integralLine integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
of
over asBy the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
and the fundamental theorem of calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
one then has
Therefore the integral of
over does not depend on the actual path , but only on its endpoints, which is what we wanted to show.Sufficiency
Next we show that if g is holomorphic, and the integral of over any path depends only on the endpoints, then g has an antiderivative. We will do so by finding an anti-derivative explicitly.Without loss of generality, we can assume that the domain
of is connected, as otherwise one can prove the existence of an antiderivative on each connected component. With this assumption, fix a point in and for any in define the function
where
is any path joining to . Such a path exists since is assumed to be an open connected set. The function is well-defined because the integral depends only on the endpoints of .That this
is an antiderivative of can be argued in the same way as the real case. We have, for a given z in U,where [z, w] denotes the line segment between z and w. By continuity of g, the final expression goes to zero as w approaches z. In other words, f' = g.