In the mathematical field of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, contour integration is a method of evaluating certain
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s along paths in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
.
Contour integration is closely related to the
calculus of residues,
a method of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
.
One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods.
Contour integration methods include:
* direct integration of a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued function along a curve in the complex plane (a ''
contour
Contour may refer to:
* Contour (linguistics), a phonetic sound
* Pitch contour
* Contour (camera system), a 3D digital camera system
* Contour, the KDE Plasma 4 interface for tablet devices
* Contour line, a curve along which the function ha ...
'');
* application of the
Cauchy integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
; and
* application of the
residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
.
One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums.
Curves in the complex plane
In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
a contour is a type of curve in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. In contour integration, contours provide a precise definition of the
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
s on which an integral may be suitably defined. A curve in the complex plane is defined as a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
from a
closed interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
to the complex plane: .
This definition of a curve coincides with the intuitive notion of a curve, but includes a parametrization by a continuous function from a closed interval. This more precise definition allows us to consider what properties a curve must have for it to be useful for integration. In the following subsections we narrow down the set of curves that we can integrate to include only those that can be built up out of a finite number of continuous curves that can be given a direction. Moreover, we will restrict the "pieces" from crossing over themselves, and we require that each piece have a finite (non-vanishing) continuous derivative. These requirements correspond to requiring that we consider only curves that can be traced, such as by a pen, in a sequence of even, steady strokes, which stop only to start a new piece of the curve, all without picking up the pen.
Directed smooth curves
Contours are often defined in terms of directed smooth curves.
[ These provide a precise definition of a "piece" of a smooth curve, of which a contour is made.
A smooth curve is a curve with a non-vanishing, continuous derivative such that each point is traversed only once ( is one-to-one), with the possible exception of a curve such that the endpoints match (). In the case where the endpoints match the curve is called closed, and the function is required to be one-to-one everywhere else and the derivative must be continuous at the identified point (). A smooth curve that is not closed is often referred to as a smooth arc.][
The parametrization of a curve provides a natural ordering of points on the curve: comes before if . This leads to the notion of a directed smooth curve. It is most useful to consider curves independent of the specific parametrization. This can be done by considering ]equivalence classes
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of smooth curves with the same direction. A directed smooth curve can then be defined as an ordered set of points in the complex plane that is the image of some smooth curve in their natural order (according to the parametrization). Note that not all orderings of the points are the natural ordering of a smooth curve. In fact, a given smooth curve has only two such orderings. Also, a single closed curve can have any point as its endpoint, while a smooth arc has only two choices for its endpoints.
Contours
Contours are the class of curves on which we define contour integration. A contour is a directed curve which is made up of a finite sequence of directed smooth curves whose endpoints are matched to give a single direction. This requires that the sequence of curves be such that the terminal point of coincides with the initial point of , . This includes all directed smooth curves. Also, a single point in the complex plane is considered a contour. The symbol + is often used to denote the piecing of curves together to form a new curve. Thus we could write a contour that is made up of curves as
Contour integrals
The contour integral of a complex function
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 helpful in many branches of mathematics, including algebraic ...
is a generalization of the integral for real-valued functions. For continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s in the complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, the contour integral can be defined in analogy to the line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
by first defining the integral along a directed smooth curve in terms of an integral over a real valued parameter. A more general definition can be given in terms of partitions of the contour in analogy with the partition of an interval
In mathematics, a partition of an interval on the real line is a finite sequence of real numbers such that
:.
In other terms, a partition of a compact interval is a strictly increasing sequence of numbers (belonging to the interval itself) ...
and the Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
. In both cases the integral over a contour is defined as the sum of the integrals over the directed smooth curves that make up the contour.
For continuous functions
To define the contour integral in this way one must first consider the integral, over a real variable, of a complex-valued function. Let be a complex-valued function of a real variable, . The real and imaginary parts of are often denoted as and , respectively, so that
Then the integral of the complex-valued function over the interval is given by
Let be a continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
on the directed smooth curve . Let be any parametrization of that is consistent with its order (direction). Then the integral along is denoted
and is given by[
This definition is well defined. That is, the result is independent of the parametrization chosen.] In the case where the real integral on the right side does not exist the integral along is said not to exist.
As a generalization of the Riemann integral
The generalization of the Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
to functions of a complex variable is done in complete analogy to its definition for functions from the real numbers. The partition of a directed smooth curve is defined as a finite, ordered set of points on . The integral over the curve is the limit of finite sums of function values, taken at the points on the partition, in the limit that the maximum distance between any two successive points on the partition (in the two-dimensional complex plane), also known as the mesh, goes to zero.
Direct methods
Direct methods involve the calculation of the integral by means of methods similar to those in calculating line integrals in multivariate calculus. This means that we use the following method:
*parametrizing the contour
*: The contour is parametrized by a differentiable complex-valued function of real variables, or the contour is broken up into pieces and parametrized separately.
* substitution of the parametrization into the integrand
*: Substituting the parametrization into the integrand transforms the integral into an integral of one real variable.
* direct evaluation
*: The integral is evaluated in a method akin to a real-variable integral.
Example
A fundamental result in complex analysis is that the contour integral of is , where the path of the contour is taken to be the unit circle traversed counterclockwise (or any positively oriented Jordan curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
about 0). In the case of the unit circle there is a direct method to evaluate the integral
In evaluating this integral, use the unit circle as a contour, parametrized by , with , then and
which is the value of the integral.
Applications of integral theorems
Applications of integral theorems are also often used to evaluate the contour integral along a contour, which means that the real-valued integral is calculated simultaneously along with calculating the contour integral.
Integral theorems such as the Cauchy integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
or residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
are generally used in the following method:
* a specific contour is chosen:
*: The contour is chosen so that the contour follows the part of the complex plane that describes the real-valued integral, and also encloses singularities of the integrand so application of the Cauchy integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
or residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
is possible
* application of Cauchy's integral theorem
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in t ...
*: The integral is reduced to only an integration around a small circle about each pole.
* application of the Cauchy integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
or residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
*: Application of these integral formulae gives us a value for the integral around the whole of the contour.
* division of the contour into a contour along the real part and imaginary part
*: The whole of the contour can be divided into the contour that follows the part of the complex plane that describes the real-valued integral as chosen before (call it ), and the integral that crosses the complex plane (call it ). The integral over the whole of the contour is the sum of the integral over each of these contours.
* demonstration that the integral that crosses the complex plane plays no part in the sum
*: If the integral can be shown to be zero, or if the real-valued integral that is sought is improper, then if we demonstrate that the integral as described above tends to 0, the integral along will tend to the integral around the contour .
* conclusion
*: If we can show the above step, then we can directly calculate , the real-valued integral.
Example 1
Consider the integral
To evaluate this integral, we look at the complex-valued function
which has singularities at and . We choose a contour that will enclose the real-valued integral, here a semicircle with boundary diameter on the real line (going from, say, to ) will be convenient. Call this contour .
There are two ways of proceeding, using the Cauchy integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
or by the method of residues:
Using the Cauchy integral formula
Note that:
thus
Furthermore, observe that
Since the only singularity in the contour is the one at , then we can write
which puts the function in the form for direct application of the formula. Then, by using Cauchy's integral formula,
We take the first derivative, in the above steps, because the pole is a second-order pole. That is, is taken to the second power, so we employ the first derivative of . If it were taken to the third power, we would use the second derivative and divide by 2!, etc. The case of to the first power corresponds to a zero order derivative—just itself.
We need to show that the integral over the arc of the semicircle tends to zero as , using the estimation lemma In mathematics the estimation lemma, also known as the inequality, gives an upper bound for a contour integral. If is a complex-valued, continuous function on the contour and if its absolute value is bounded by a constant for all on , then
...
where is an upper bound on along the arc and the length of the arc. Now,
So
Using the method of residues
Consider the Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
of about , the only singularity we need to consider. We then have
(See the sample Laurent calculation from Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
for the derivation of this series.)
It is clear by inspection that the residue is , so, by the residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
, we have
Thus we get the same result as before.
Contour note
As an aside, a question can arise whether we do not take the semicircle to include the ''other'' singularity, enclosing . To have the integral along the real axis moving in the correct direction, the contour must travel clockwise, i.e., in a negative direction, reversing the sign of the integral overall.
This does not affect the use of the method of residues by series.
Example 2 – Cauchy distribution
The integral
(which arises in probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
as a scalar multiple of the characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of the Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
) resists the techniques of elementary calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. We will evaluate it by expressing it as a limit of contour integrals along the contour that goes along the real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
line from to and then counterclockwise along a semicircle centered at 0 from to . Take to be greater than 1, so that the imaginary unit is enclosed within the curve. The contour integral is
Since is an entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
(having no singularities at any point in the complex plane), this function has singularities only where the denominator is zero. Since , that happens only where or . Only one of those points is in the region bounded by this contour. The residue
Residue may refer to:
Chemistry and biology
* An amino acid, within a peptide chain
* Crop residue, materials left after agricultural processes
* Pesticide residue, refers to the pesticides that may remain on or in food after they are applied ...
of at is
According to the residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
, then, we have
The contour may be split into a "straight" part and a curved arc, so that
and thus
According to Jordan's lemma In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. The lemma is named after the French mathematician Camille Jordan.
Statement
Consider a ...
, if then
Therefore, if then
A similar argument with an arc that winds around rather than shows that if then
and finally we have this:
(If then the integral yields immediately to real-valued calculus methods and its value is .)
Example 3 – trigonometric integrals
Certain substitutions can be made to integrals involving trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
, so the integral is transformed into a rational function of a complex variable and then the above methods can be used in order to evaluate the integral.
As an example, consider
We seek to make a substitution of . Now, recall
and
Taking to be the unit circle, we substitute to get:
The singularities to be considered are at Let be a small circle about and be a small circle about Then we arrive at the following:
Example 3a – trigonometric integrals, the general procedure
The above method may be applied to all integrals of the type
where and are polynomials, i.e. a rational function in trigonometric terms is being integrated. Note that the bounds of integration may as well be and −, as in the previous example, or any other pair of endpoints 2 apart.
The trick is to use the substitution where and hence
This substitution maps the interval to the unit circle. Furthermore,
and
so that a rational function in results from the substitution, and the integral becomes
which is in turn computed by summing the residues of inside the unit circle.
The image at right illustrates this for
which we now compute. The first step is to recognize that
The substitution yields
The poles of this function are at and . Of these, and are outside the unit circle (shown in red, not to scale), whereas and are inside the unit circle (shown in blue). The corresponding residues are both equal to , so that the value of the integral is
Example 4 – branch cuts
Consider the real integral
We can begin by formulating the complex integral
We can use the Cauchy integral formula or residue theorem again to obtain the relevant residues. However, the important thing to note is that , so has a branch cut
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
. This affects our choice of the contour . Normally the logarithm branch cut is defined as the negative real axis, however, this makes the calculation of the integral slightly more complicated, so we define it to be the positive real axis.
Then, we use the so-called ''keyhole contour'', which consists of a small circle about the origin of radius say, extending to a line segment parallel and close to the positive real axis but not touching it, to an almost full circle, returning to a line segment parallel, close, and below the positive real axis in the negative sense, returning to the small circle in the middle.
Note that and are inside the big circle. These are the two remaining poles, derivable by factoring the denominator of the integrand. The branch point at was avoided by detouring around the origin.
Let be the small circle of radius , the larger, with radius , then
It can be shown that the integrals over and both tend to zero as and , by an estimation argument above, that leaves two terms. Now since , on the contour outside the branch cut, we have gained 2 in argument along . (By Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality
e^ + 1 = 0
where
: is Euler's number, the base of natural logarithms,
: is the imaginary unit, which by definition satisfies , and
: is pi, the ratio of the circ ...
, represents the unit vector, which therefore has as its log. This is what is meant by the argument of . The coefficient of forces us to use 2.) So
Therefore:
By using the residue theorem or the Cauchy integral formula (first employing the partial fractions method to derive a sum of two simple contour integrals) one obtains
Example 5 – the square of the logarithm
This section treats a type of integral of which
is an example.
To calculate this integral, one uses the function
and the branch of the logarithm corresponding to .
We will calculate the integral of along the keyhole contour shown at right. As it turns out this integral is a multiple of the initial integral that we wish to calculate and by the Cauchy residue theorem we have
Let be the radius of the large circle, and the radius of the small one. We will denote the upper line by , and the lower line by . As before we take the limit when and . The contributions from the two circles vanish. For example, one has the following upper bound with the lemma:
In order to compute the contributions of and we set on and on , with :
which gives
Example 6 – logarithms and the residue at infinity
We seek to evaluate
This requires a close study of
We will construct so that it has a branch cut on , shown in red in the diagram. To do this, we choose two branches of the logarithm, setting
and
The cut of is therefore and the cut of is . It is easy to see that the cut of the product of the two, i.e. , is , because is actually continuous across . This is because when and we approach the cut from above, has the value
When we approach from below, has the value
But
so that we have continuity across the cut. This is illustrated in the diagram, where the two black oriented circles are labelled with the corresponding value of the argument of the logarithm used in and .
We will use the contour shown in green in the diagram. To do this we must compute the value of along the line segments just above and just below the cut.
Let (in the limit, i.e. as the two green circles shrink to radius zero), where . Along the upper segment, we find that has the value
and along the lower segment,
It follows that the integral of along the upper segment is in the limit, and along the lower segment, .
If we can show that the integrals along the two green circles vanish in the limit, then we also have the value of , by the Cauchy residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
. Let the radius of the green circles be , where and , and apply the inequality. For the circle on the left, we find
Similarly, for the circle on the right, we have
Now using the Cauchy residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
, we have
where the minus sign is due to the clockwise direction around the residues. Using the branch of the logarithm from before, clearly
The pole is shown in blue in the diagram. The value simplifies to
We use the following formula for the residue at infinity:
Substituting, we find
and
where we have used the fact that for the second branch of the logarithm. Next we apply the binomial expansion, obtaining
The conclusion is that
Finally, it follows that the value of is
which yields
Evaluation with residue theorem
Using the residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
, we can evaluate closed contour integrals. The following are examples on evaluating contour integrals with the residue theorem.
Using the residue theorem, let's evaluate this contour integral.
As a refresher, the residue theorem states
where is the residue of .
has only one pole, . From that, we can determine the residue
Residue may refer to:
Chemistry and biology
* An amino acid, within a peptide chain
* Crop residue, materials left after agricultural processes
* Pesticide residue, refers to the pesticides that may remain on or in food after they are applied ...
of to be
Thus, using the residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
, we can determine:
Multivariable contour integrals
To solve multivariable contour integrals (i.e. surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
s, complex volume integrals, and higher order integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s), we must use the divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
. For right now, let be interchangeable with . These will both serve as the divergence of the vector field denoted as . This theorem states:
In addition, we also need to evaluate where is an alternate notation of . The divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of any dimension can be described as
Example 1
Let the vector field and be bounded by the following
The corresponding double contour integral would be set up as such:
We now evaluate . While we're at it, let's set up the corresponding triple integral:
Example 2
Let the vector field , and remark that there are 4 parameters in this case. Let this vector field be bounded by the following:
To evaluate this, we must utilize the divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
as stated before, and we must evaluate . Let
Thus, we can evaluate a contour integral with . We can use the same method to evaluate contour integrals for any vector field with as well.
Integral representation
An integral representation of a function is an expression of the function involving a contour integral. Various integral representations are known for many special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by ...
s. Integral representations can be important for theoretical reasons, e.g. giving analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
or functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
s, or sometimes for numerical evaluations.
For example, the original definition of the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
via a Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analyti ...
,
is valid only for . But
where the integration is done over the Hankel contour
In mathematics, a Hankel contour is a path in the complex plane which extends from
(+∞,δ), around the origin counter clockwise and back to
(+∞,−δ), where δ is an arbitrarily small positive number. The contour thus remains arbitrarily ...
, is valid for all complex s not equal to 1.
See also
* Residue (complex analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for an ...
* Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
Formulation
Depending on the type of singularity in the integrand , ...
* Poisson integral
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriv ...
* Pochhammer contour
In mathematics, the Pochhammer contour, introduced by Jordan (1887), pp. 243–244 and , is a contour in the complex plane with two points removed, used for contour integration. If ''A'' and ''B'' are loops around the two points, both starting ...
References
Further reading
*
* Jean Jacquelin, Marko Riedel,
Branche univalente
Les-Mathematiques.net'', in French.
* Marko Riedel et al.,
Problème d'intégrale
Les-Mathematiques.net'', in French.
* Marko Riedel et al.,
Integral by residue
math.stackexchange.com''.
* W W L Chen,
Introduction to Complex Analysis
'
* Various authors,
sin límites ni cotas
es.ciencia.matematicas'', in Spanish.
External links
*
{{DEFAULTSORT:Methods Of Contour Integration
Complex analysis