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Contour integration

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In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.<ref name=Stalker>Template:Cite book</ref><ref name=Bak>Template:Cite book</ref><ref name=Krantz>Template:Cite book</ref>

Contour integration is closely related to the calculus of residues,<ref name=Mitrinovic1>Template:Cite book</ref> a method of complex analysis.

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. It also has various applications in physics.<ref name=Mitrinovic2>Template:Cite book</ref>

Contour integration methods include:

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 planeEdit

In complex analysis, a contour is a type of curve in the complex plane. In contour integration, contours provide a precise definition of the curves on which an integral may be suitably defined. A curve in the complex plane is defined as a continuous function from a closed interval of the real line to the complex plane: <math>z:[a,b]\to\C</math>.

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.<ref name=Saff>Template:Cite book</ref>

Directed smooth curvesEdit

Contours are often defined in terms of directed smooth curves.<ref name=Saff/> These provide a precise definition of a "piece" of a smooth curve, of which a contour is made.

A smooth curve is a curve <math>z:[a,b]\to\C</math> with a non-vanishing, continuous derivative such that each point is traversed only once (Template:Mvar is one-to-one), with the possible exception of a curve such that the endpoints match (<math>z(a)=z(b)</math>). 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 (<math>z'(a)=z'(b)</math>). A smooth curve that is not closed is often referred to as a smooth arc.<ref name=Saff/>

The parametrization of a curve provides a natural ordering of points on the curve: <math>z(x)</math> comes before <math>z(y)</math> if <math>x<y</math>. 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 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.

ContoursEdit

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 <math>\gamma_1,\dots,\gamma_n</math> be such that the terminal point of <math>\gamma_i</math> coincides with the initial point of <math>\gamma_{i+1}</math> for all <math>i</math> such that <math>1\leq i<n</math> . This includes all directed smooth curves. Also, a single point in the complex plane is considered a contour. The symbol <math>+</math> is often used to denote the piecing of curves together to form a new curve. Thus we could write a contour <math>\Gamma</math> that is made up of <math>n</math> curves as <math display=block> \Gamma = \gamma_1 + \gamma_2 + \cdots + \gamma_n.</math>

Contour integralsEdit

The contour integral of a complex function <math>f:\C\to\C</math> is a generalization of the integral for real-valued functions. For continuous functions in the complex plane, the contour integral can be defined in analogy to the line integral 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 and the Riemann integral. 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 functionsEdit

To define the contour integral in this way one must first consider the integral, over a real variable, of a complex-valued function. Let <math>f:\R\to\C</math> be a complex-valued function of a real variable, <math>t</math>. The real and imaginary parts of <math>f</math> are often denoted as <math>u(t)</math> and <math>v(t)</math>, respectively, so that <math display=block>f(t) = u(t) + iv(t).</math> Then the integral of the complex-valued function <math>f</math> over the interval <math>[a,b]</math> is given by <math display=block>\begin{align} \int_a^b f(t) \, dt &= \int_a^b \big( u(t) + i v(t) \big) \, dt \\ &= \int_a^b u(t) \, dt + i \int_a^b v(t) \, dt. \end{align}</math>

Now, to define the contour integral, let <math>f:\C\to\C</math> be a continuous function on the directed smooth curve <math>\gamma</math>. Let <math>z:[a,b]\to\C</math> be any parametrization of <math>\gamma</math> that is consistent with its order (direction). Then the integral along <math>\gamma</math> is denoted <math display=block>\int_\gamma f(z)\, dz\, </math> and is given by<ref name=Saff/> <math display="block">\int_\gamma f(z) \, dz := \int_a^b f\big(z(t)\big) z'(t) \, dt.</math>

This definition is well defined. That is, the result is independent of the parametrization chosen.<ref name="Saff" /> In the case where the real integral on the right side does not exist the integral along <math>\gamma</math> is said not to exist.

As a generalization of the Riemann integralEdit

The generalization of the Riemann integral 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 <math>\gamma</math> is defined as a finite, ordered set of points on <math>\gamma</math>. 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 methodsEdit

Direct methods involve the calculation of the integral through 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.

ExampleEdit

A fundamental result in complex analysis is that the contour integral of Template:Math is Template:Math, where the path of the contour is taken to be the unit circle traversed counterclockwise (or any positively oriented Jordan curve about 0). In the case of the unit circle there is a direct method to evaluate the integral <math display=block>\oint_C \frac{1}{z}\,dz.</math>

In evaluating this integral, use the unit circle Template:Math as a contour, parametrized by Template:Math, with Template:Math, then Template:Math and <math display=block>\oint_C \frac{1}{z}\,dz = \int_0^{2\pi} \frac{1}{e^{it}} ie^{it}\,dt = i\int_0^{2\pi} 1 \, dt = i \, t\Big|_0^{2\pi} = \left(2\pi-0\right)i = 2\pi i</math>

which is the value of the integral. This result only applies to the case in which z is raised to power of -1. If the power is not equal to -1, then the result will always be zero.

Applications of integral theoremsEdit

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 or residue theorem 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 or residue theorem is possible
  • application of Cauchy's integral theorem
    The integral is reduced to only an integration around a small circle about each pole.
  • application of the Cauchy integral formula or residue theorem
    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 Template:Mvar), and the integral that crosses the complex plane (call it Template:Mvar). 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 Template:Mvar can be shown to be zero, or if the real-valued integral that is sought is improper, then if we demonstrate that the integral Template:Mvar as described above tends to 0, the integral along Template:Mvar will tend to the integral around the contour Template:Math.
  • conclusion
    If we can show the above step, then we can directly calculate Template:Mvar, the real-valued integral.

Example 1Edit

Consider the integral <math display=block>\int_{-\infty}^\infty \frac{1}{\left(x^2+1\right)^2}\,dx,</math>

To evaluate this integral, we look at the complex-valued function <math display=block>f(z)=\frac{1}{\left(z^2+1\right)^2}</math>

which has singularities at Template:Mvar and Template:Math. We choose a contour that will enclose the real-valued integral, here a semicircle with boundary diameter on the real line (going from, say, Template:Math to Template:Mvar) will be convenient. Call this contour Template:Mvar.

There are two ways of proceeding, using the Cauchy integral formula or by the method of residues:

Using the Cauchy integral formulaEdit

Note that: <math display=block>\oint_C f(z)\,dz = \int_{-a}^a f(z)\,dz + \int_\text{Arc} f(z)\,dz </math> thus <math display=block>\int_{-a}^a f(z)\,dz = \oint_C f(z)\,dz - \int_\text{Arc} f(z)\,dz </math>

Furthermore, observe that <math display=block>f(z)=\frac{1}{\left(z^2+1\right)^2}=\frac{1}{(z+i)^2(z-i)^2}.</math>

Since the only singularity in the contour is the one at Template:Mvar, then we can write <math display=block>f(z)=\frac{\frac{1}{(z+i)^2}}{(z-i)^2},</math>

which puts the function in the form for direct application of the formula. Then, by using Cauchy's integral formula, <math display=block>\oint_C f(z)\,dz = \oint_C \frac{\frac{1}{(z+i)^2}}{(z-i)^2}\,dz = 2\pi i \, \left.\frac{d}{dz} \frac{1}{(z+i)^2}\right|_{z=i} =2 \pi i \left[\frac{-2}{(z+i)^3}\right]_{z = i} =\frac{\pi}{2}</math>

We take the first derivative, in the above steps, because the pole is a second-order pole. That is, Template:Math is taken to the second power, so we employ the first derivative of Template:Math. If it were Template:Math taken to the third power, we would use the second derivative and divide by Template:Math, etc. The case of Template:Math to the first power corresponds to a zero order derivative—just Template:Math itself.

We need to show that the integral over the arc of the semicircle tends to zero as Template:Math, using the estimation lemma <math display=block>\left|\int_\text{Arc} f(z)\,dz\right| \le ML</math>

where Template:Mvar is an upper bound on Template:Math along the arc and Template:Mvar the length of the arc. Now, <math display=block>\left|\int_\text{Arc} f(z)\,dz\right|\le \frac{a\pi}{\left(a^2-1\right)^2} \to 0 \text{ as } a \to \infty.</math> So <math display=block>\int_{-\infty}^\infty \frac{1}{\left(x^2+1\right)^2}\,dx = \int_{-\infty}^\infty f(z)\,dz = \lim_{a \to +\infty} \int_{-a}^a f(z)\,dz = \frac{\pi}2.\quad\square</math>

Using the method of residuesEdit

Consider the Laurent series of Template:Math about Template:Mvar, the only singularity we need to consider. We then have <math display=block>f(z) = \frac{-1}{4(z-i)^2} + \frac{-i}{4(z-i)} + \frac{3}{16} + \frac{i}{8}(z-i) + \frac{-5}{64}(z-i)^2 + \cdots</math>

(See the sample Laurent calculation from Laurent series for the derivation of this series.)

It is clear by inspection that the residue is Template:Math, so, by the residue theorem, we have <math display=block>\oint_C f(z)\,dz = \oint_C \frac{1}{\left(z^2+1\right)^2}\,dz = 2 \pi i \,\operatorname{Res}_{z=i} f(z) = 2 \pi i \left(-\frac{i}{4}\right)=\frac{\pi}2 \quad\square</math>

Thus we get the same result as before.

Contour noteEdit

As an aside, a question can arise whether we do not take the semicircle to include the other singularity, enclosing Template:Math. 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 distributionEdit

The integral <math display=block>\int_{-\infty}^\infty \frac{e^{itx}}{x^2+1}\,dx</math>

the contour

(which arises in probability theory as a scalar multiple of the characteristic function of the Cauchy distribution) resists the techniques of elementary calculus. We will evaluate it by expressing it as a limit of contour integrals along the contour Template:Mvar that goes along the real line from Template:Math to Template:Mvar and then counterclockwise along a semicircle centered at 0 from Template:Mvar to Template:Math. Take Template:Mvar to be greater than 1, so that the imaginary unit Template:Mvar is enclosed within the curve. The contour integral is <math display=block>\int_C \frac{e^{itz} }{ z^2+1}\,dz.</math>

Since Template:Math is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the denominator Template:Math is zero. Since Template:Math, that happens only where Template:Math or Template:Math. Only one of those points is in the region bounded by this contour. The residue of Template:Math at Template:Math is <math display=block>\lim_{z\to i}(z-i)f(z) = \lim_{z\to i}(z-i)\frac{e^{itz} }{ z^2+1} = \lim_{z\to i}(z-i)\frac{e^{itz} }{ (z-i)(z+i)} = \lim_{z\to i}\frac{e^{itz} }{ z+i} = \frac{e^{-t}}{2i}.</math>

According to the residue theorem, then, we have <math display=block>\int_C f(z)\,dz=2\pi i \operatorname{Res}_{z=i}f(z)=2\pi i\frac{e^{-t} }{ 2i}=\pi e^{-t}.</math>

The contour Template:Mvar may be split into a "straight" part and a curved arc, so that <math display=block>\int_\text{straight}+\int_\text{arc}=\pi e^{-t},</math> and thus <math display=block>\int_{-a}^a =\pi e^{-t}-\int_\text{arc}.</math>

According to Jordan's lemma, if Template:Math then <math display=block>\int_\text{arc}\frac{e^{itz} }{ z^2+1}\,dz \rightarrow 0 \mbox{ as } a\rightarrow\infty.</math>

Therefore, if Template:Math then <math display=block>\int_{-\infty}^\infty \frac{e^{itx} }{ x^2+1}\,dx=\pi e^{-t}.</math>

A similar argument with an arc that winds around Template:Math rather than Template:Mvar shows that if Template:Math then <math display=block>\int_{-\infty}^\infty \frac{e^{itx} }{ x^2+1}\,dx=\pi e^t,</math> and finally we have this: <math display=block>\int_{-\infty}^\infty \frac{e^{itx} }{ x^2+1} \,dx=\pi e^{-|t|}.</math>

(If Template:Math then the integral yields immediately to real-valued calculus methods and its value is Template:Pi.)

Example 3 – trigonometric integralsEdit

Certain substitutions can be made to integrals involving trigonometric functions, 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 <math display=block>\int_{-\pi}^\pi \frac{1 }{ 1 + 3 (\cos t)^2} \,dt.</math>

We seek to make a substitution of Template:Math. Now, recall <math display=block>\cos t = \frac12 \left(e^{it}+e^{-it}\right) = \frac12 \left(z+\frac{1}{z}\right)</math> and <math display=block>\frac{dz}{dt} = iz,\ dt = \frac{dz}{iz}.</math>

Taking Template:Mvar to be the unit circle, we substitute to get:

<math>\begin{align} \oint_C \frac{1}{ 1 + 3 \left(\frac12 \left(z+\frac{1}{z}\right)\right)^2} \,\frac{dz}{iz} &= \oint_C \frac{1 }{ 1 + \frac34 \left(z+\frac{1}{z}\right)^2}\frac{1}{iz} \,dz \\ &= \oint_C \frac{-i}{ z+\frac34 z\left(z+\frac{1}{z}\right)^2}\,dz \\ &= -i \oint_C \frac{dz}{ z+\frac34 z\left(z^2+2+\frac{1}{z^2}\right)} \\ &= -i \oint_C \frac{dz}{ z+\frac34 \left(z^3+2z+\frac{1}{z}\right)} \\ &= -i \oint_C \frac{dz}{ \frac34 z^3+\frac52 z+\frac{3}{4z}} \\ &= -i \oint_C \frac{4}{ 3z^3+10z+\frac{3}{z}}\,dz \\ &= -4i \oint_C \frac{dz}{ 3z^3+10z+\frac{3}{z}} \\ &= -4i \oint_C \frac{z}{ 3z^4+10z^2+3 } \,dz \\ &= -4i \oint_C \frac{z}{ 3\left(z+\sqrt{3}i\right)\left(z-\sqrt{3}i\right)\left(z+\frac{i}{\sqrt 3}\right)\left(z-\frac{i}{\sqrt 3}\right)}\,dz \\ &= -\frac{4i}{3} \oint_C \frac{z}{\left(z+\sqrt{3}i\right)\left(z-\sqrt{3}i\right)\left(z+\frac{i}{\sqrt 3}\right)\left(z-\frac{i}{\sqrt 3}\right)}\,dz. \end{align}</math>

The singularities to be considered are at <math>\tfrac{\pm i}{\sqrt{3}}.</math> Let Template:Math be a small circle about <math>\tfrac{i}{\sqrt{3}},</math> and Template:Math be a small circle about <math>\tfrac{-i}{\sqrt{3}}.</math> Then we arrive at the following: <math display=block>\begin{align} & -\frac{4i}{3} \left [\oint_{C_1} \frac{\frac{z}{\left(z+\sqrt{3}i\right)\left(z-\sqrt{3}i\right)\left(z+\frac{i}{\sqrt 3} \right)}}{z-\frac{i}{\sqrt 3}}\,dz +\oint_{C_2} \frac{\frac{z}{\left(z+\sqrt{3}i\right)\left(z-\sqrt{3}i\right)\left(z-\frac{i}{\sqrt 3}\right)}}{z+\frac{i}{\sqrt 3}} \, dz \right ] \\ = {} & -\frac{4i}{3} \left[ 2\pi i \left[\frac{z}{\left(z+\sqrt{3}i\right)\left(z-\sqrt{3}i\right)\left(z+\frac{i}{\sqrt 3}\right)}\right]_{z=\frac{i}{\sqrt 3}} + 2\pi i \left[\frac{z}{\left(z+\sqrt{3}i\right)\left(z-\sqrt{3}i\right)\left(z-\frac{i}{\sqrt 3}\right)} \right]_{z=-\frac{i}{\sqrt 3}}\right] \\ = {} & \frac{8\pi}{3} \left[\frac{\frac{i}{\sqrt 3}}{\left(\frac{i}{\sqrt 3}+\sqrt{3}i\right)\left(\frac{i}{\sqrt 3}-\sqrt{3}i\right)\left(\frac{i}{\sqrt 3}+\frac{i}{\sqrt 3}\right)} + \frac{-\frac{i}{\sqrt 3}}{\left(-\frac{i}{\sqrt 3}+\sqrt{3}i\right)\left(-\frac{i}{\sqrt 3}-\sqrt{3}i\right)\left(-\frac{i}{\sqrt 3}-\frac{i}{\sqrt 3}\right)} \right] \\ = {} & \frac{8\pi}{3} \left[\frac{\frac{i}{\sqrt 3}}{\left(\frac{4}{\sqrt 3}i\right)\left(-\frac{2}{i\sqrt{3}}\right)\left(\frac{2}{\sqrt{3}i}\right)}+\frac{-\frac{i}{\sqrt 3}}{\left(\frac{2}{\sqrt 3}i\right)\left(-\frac{4}{\sqrt 3}i\right)\left(-\frac{2}{\sqrt 3}i\right)}\right] \\ = {} & \frac{8\pi}{3}\left[\frac{\frac{i}{\sqrt 3}}{i\left(\frac{4}{\sqrt 3}\right)\left(\frac{2}{\sqrt 3}\right)\left(\frac{2}{\sqrt 3}\right)}+\frac{-\frac{i}{\sqrt 3}}{-i\left(\frac{2}{\sqrt 3}\right)\left(\frac{4}{\sqrt 3}\right)\left(\frac{2}{\sqrt 3}\right)}\right] \\ = {} & \frac{8\pi}{3}\left[\frac{\frac{1}{\sqrt 3}}{\left(\frac{4}{\sqrt 3}\right)\left(\frac{2}{\sqrt 3}\right)\left(\frac{2}{\sqrt 3}\right)}+\frac{\frac{1}{\sqrt 3}}{\left(\frac{2}{\sqrt 3}\right)\left(\frac{4}{\sqrt 3}\right)\left(\frac{2}{\sqrt 3}\right)}\right] \\ = {} & \frac{8\pi}{3}\left[\frac{\frac{1}{\sqrt 3}}{\frac{16}{3\sqrt{3}}}+\frac{\frac{1}{\sqrt 3}}{\frac{16}{3\sqrt{3}}} \right] \\ = {} & \frac{8\pi}{3}\left[\frac{3}{16} + \frac{3}{16} \right] \\ = {} & \pi. \end{align}</math>

Example 3a – trigonometric integrals, the general procedureEdit

The above method may be applied to all integrals of the type <math display=block> \int_0^{2\pi} \frac{P\big(\sin(t),\sin(2t),\ldots,\cos(t),\cos(2t),\ldots\big)}{Q\big(\sin(t),\sin(2t),\ldots,\cos(t),\cos(2t),\ldots\big)}\, dt</math>

where Template:Mvar and Template:Mvar are polynomials, i.e. a rational function in trigonometric terms is being integrated. Note that the bounds of integration may as well be Template:Pi and −Template:Pi, as in the previous example, or any other pair of endpoints 2Template:Pi apart.

The trick is to use the substitution Template:Math where Template:Math and hence <math display=block> \frac{1}{iz} \,dz = dt.</math>

This substitution maps the interval Template:Closed-closed to the unit circle. Furthermore, <math display=block> \sin(k t) = \frac{e^{i k t} - e^{- i k t}}{2 i} = \frac{z^k - z^{-k}}{2i}</math> and <math display=block> \cos(k t) = \frac{e^{i k t} + e^{- i k t}}{2} = \frac{z^k + z^{-k}}{2}</math> so that a rational function Template:Math in Template:Mvar results from the substitution, and the integral becomes <math display=block> \oint_{|z|=1} f(z) \frac{1}{iz}\, dz </math> which is in turn computed by summing the residues of Template:Math inside the unit circle.

File:TrigonometricToComplex.png

The image at right illustrates this for <math display=block> I = \int_0^\frac{\pi}{2} \frac{1}{1 + (\sin t)^2}\, dt,</math> which we now compute. The first step is to recognize that <math display=block> I = \frac14 \int_0^{2\pi} \frac{1}{1 + (\sin t)^2} \,dt.</math>

The substitution yields <math display=block> \frac{1}{4} \oint_{|z|=1} \frac{4 i z}{z^4 - 6z^2 + 1}\, dz = \oint_{|z|=1} \frac{i z}{z^4 - 6z^2 + 1}\, dz.</math>

The poles of this function are at Template:Math and Template:Math. Of these, Template:Math and Template:Math are outside the unit circle (shown in red, not to scale), whereas Template:Math and Template:Math are inside the unit circle (shown in blue). The corresponding residues are both equal to Template:Math, so that the value of the integral is <math display=block> I = 2 \pi i \; 2 \left( - \frac{\sqrt{2}}{16} i \right) = \pi \frac{\sqrt{2}}{4}.</math>

Example 4 – branch cutsEdit

Consider the real integral <math display=block>\int_0^\infty \frac{\sqrt x}{x^2+6x+8}\,dx.</math>

We can begin by formulating the complex integral <math display=block>\int_C \frac{\sqrt z}{z^2+6z+8}\,dz=I.</math>

File:Keyhole contour.svg

We can use the Cauchy integral formula or residue theorem again to obtain the relevant residues. However, the important thing to note is that Template:Math, so Template:Math has a branch cut. This affects our choice of the contour Template:Mvar. 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 Template:Mvar 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 Template:Math and Template:Math are inside the big circle. These are the two remaining poles, derivable by factoring the denominator of the integrand. The branch point at Template:Math was avoided by detouring around the origin.

Let Template:Mvar be the small circle of radius Template:Mvar, Template:Math the larger, with radius Template:Mvar, then <math display=block>\int_C = \int_\varepsilon^R + \int_\Gamma + \int_R^\varepsilon + \int_\gamma.</math>

It can be shown that the integrals over Template:Math and Template:Mvar both tend to zero as Template:Math and Template:Math, by an estimation argument above, that leaves two terms. Now since Template:Math, on the contour outside the branch cut, we have gained 2Template:Pi in argument along Template:Mvar. (By Euler's identity, Template:Math represents the unit vector, which therefore has Template:Pi as its log. This Template:Pi is what is meant by the argument of Template:Mvar. The coefficient of Template:Sfrac forces us to use 2Template:Pi.) So <math display=block>\begin{align} \int_R^\varepsilon \frac{\sqrt z}{z^2+6z+8}\,dz&=\int_R^\varepsilon \frac{e^{\frac12 \operatorname{Log} z}}{z^2+6z+8}\,dz \\[6pt] &=\int_R^\varepsilon \frac{e^{\frac12(\log|z|+i \arg{z})}}{z^2+6z+8}\,dz \\[6pt] & = \int_R^\varepsilon \frac{ e^{\frac12\log|z|}e^{\frac12(2\pi i)}}{z^2+6z+8}\,dz\\[6pt] &=\int_R^\varepsilon \frac{ e^{\frac12\log|z|}e^{\pi i}}{z^2+6z+8}\,dz \\[6pt] & = \int_R^\varepsilon \frac{-\sqrt z}{z^2+6z+8}\,dz\\[6pt] &=\int_\varepsilon^R \frac{\sqrt z}{z^2+6z+8}\,dz. \end{align}</math>

Therefore: <math display=block>\int_C \frac{\sqrt z}{z^2+6z+8}\,dz=2\int_0^\infty \frac{\sqrt x}{x^2+6x+8}\,dx.</math>

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 <math display=block>\pi i \left(\frac{i}{\sqrt 2}-i\right)=\int_0^\infty \frac{\sqrt x}{x^2+6x+8}\,dx = \pi\left(1-\frac{1}{\sqrt 2}\right).\quad\square</math>

Example 5 – the square of the logarithmEdit

File:KeyholeContourLeftTikz.tif

This section treats a type of integral of which <math display=block>\int_0^\infty \frac{\log x}{\left(1+x^2\right)^2} \, dx</math> is an example.

To calculate this integral, one uses the function <math display=block>f(z) = \left (\frac{\log z}{1+z^2} \right )^2</math> and the branch of the logarithm corresponding to Template:Math.

We will calculate the integral of Template:Math 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

<math>\begin{align} \left( \int_R + \int_M + \int_N + \int_r \right) f(z) \, dz = &\ 2 \pi i \big( \operatorname{Res}_{z=i} f(z) + \operatorname{Res}_{z=-i} f(z) \big) \\ = &\ 2 \pi i \left( - \frac{\pi}{4} + \frac{1}{16} i \pi^2 - \frac{\pi}{4} - \frac{1}{16} i \pi^2 \right) \\ = &\ - i \pi^2. \end{align}</math>

Let Template:Mvar be the radius of the large circle, and Template:Mvar the radius of the small one. We will denote the upper line by Template:Mvar, and the lower line by Template:Mvar. As before we take the limit when Template:Math and Template:Math. The contributions from the two circles vanish. For example, one has the following upper bound with the [[ML lemma|Template:Mvar lemma]]: <math display=block>\left| \int_R f(z) \, dz \right| \le 2 \pi R \frac{(\log R)^2 + \pi^2}{\left(R^2-1\right)^2} \to 0.</math>

In order to compute the contributions of Template:Mvar and Template:Mvar we set Template:Math on Template:Mvar and Template:Math on Template:Mvar, with Template:Math:

<math>\begin{align} -i \pi^2 &= \left( \int_R + \int_M + \int_N + \int_r \right) f(z) \, dz \\[6pt] &= \left( \int_M + \int_N \right) f(z)\, dz && \int_R, \int_r \mbox{ vanish} \\[6pt] &=-\int_\infty^0 \left (\frac{\log(-x + i\varepsilon)}{1+(-x + i\varepsilon)^2} \right )^2\, dx - \int_0^\infty \left (\frac{\log(-x - i\varepsilon)}{1+(-x - i\varepsilon)^2}\right)^2 \, dx \\[6pt] &= \int_0^\infty \left (\frac{\log(-x + i\varepsilon)}{1+(-x + i\varepsilon)^2} \right )^2 \, dx - \int_0^\infty \left (\frac{\log(-x - i\varepsilon)}{1+(-x - i\varepsilon)^2} \right )^2 \, dx \\[6pt] &= \int_0^\infty \left (\frac{\log x + i\pi}{1+x^2} \right )^2 \, dx - \int_0^\infty \left (\frac{\log x - i\pi}{1+x^2} \right )^2 \, dx && \varepsilon \to 0 \\ &= \int_0^\infty \frac{(\log x + i\pi)^2 - (\log x - i\pi)^2}{\left(1+x^2\right)^2} \, dx \\[6pt] &= \int_0^\infty \frac{4 \pi i \log x}{\left(1+x^2\right)^2} \, dx \\[6pt] &= 4 \pi i \int_0^\infty \frac{\log x}{\left(1+x^2\right)^2} \, dx \end{align}</math>

which gives <math display=block>\int_0^\infty \frac{\log x}{\left(1+x^2\right)^2} \, dx = - \frac{\pi}{4}.</math>

Example 6 – logarithms and the residue at infinityEdit

File:ContourLogs.png

We seek to evaluate <math display=block>I = \int_0^3 \frac{x^\frac34 (3-x)^\frac14}{5-x}\,dx.</math>

This requires a close study of <math display=block>f(z) = z^\frac34 (3-z)^\frac14.</math>

We will construct Template:Math so that it has a branch cut on Template:Closed-closed, shown in red in the diagram. To do this, we choose two branches of the logarithm, setting <math display=block>z^\frac34 = \exp \left (\frac34 \log z \right ) \quad \mbox{where } -\pi \le \arg z < \pi </math> and <math display=block>(3-z)^\frac14 = \exp \left (\frac14 \log(3-z) \right ) \quad \mbox{where } 0 \le \arg(3-z) < 2\pi. </math>

The cut of Template:Math is therefore Template:Open-closed and the cut of Template:Math is Template:Open-closed. It is easy to see that the cut of the product of the two, i.e. Template:Math, is Template:Math, because Template:Math is actually continuous across Template:Open-open. This is because when Template:Math and we approach the cut from above, Template:Math has the value <math display=block> r^\frac34 e^{\frac34 \pi i} (3+r)^\frac14 e^{\frac24 \pi i} = r^\frac34 (3+r)^\frac14 e^{\frac54 \pi i}.</math>

When we approach from below, Template:Math has the value <math display=block> r^\frac34 e^{-\frac34 \pi i} (3+r)^\frac14 e^{\frac04 \pi i} = r^\frac34 (3+r)^\frac14 e^{-\frac34 \pi i}.</math>

But <math display=block>e^{-\frac34 \pi i} = e^{\frac54 \pi i},</math>

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 Template:Math and Template:Math.

We will use the contour shown in green in the diagram. To do this we must compute the value of Template:Math along the line segments just above and just below the cut.

Let Template:Math (in the limit, i.e. as the two green circles shrink to radius zero), where Template:Math. Along the upper segment, we find that Template:Math has the value <math display=block>r^\frac34 e^{\frac04 \pi i} (3-r)^\frac14 e^{\frac24 \pi i} = i r^\frac34 (3-r)^\frac14</math> and along the lower segment, <math display=block>r^\frac34 e^{\frac04 \pi i} (3-r)^\frac14 e^{\frac04 \pi i} = r^\frac34 (3-r)^\frac14.</math>

It follows that the integral of Template:Math along the upper segment is Template:Math in the limit, and along the lower segment, Template:Mvar.

If we can show that the integrals along the two green circles vanish in the limit, then we also have the value of Template:Math, by the Cauchy residue theorem. Let the radius of the green circles be Template:Mvar, where Template:Math and Template:Math, and apply the [[Estimation lemma|Template:Mvar inequality]]. For the circle Template:Math on the left, we find <math display=block>\left| \int_{C_\mathrm{L}} \frac{f(z)}{5-z} dz \right| \le 2 \pi \rho \frac{\rho^\frac34 3.001^\frac14}{4.999} \in \mathcal{O} \left( \rho^\frac74 \right) \to 0.</math>

Similarly, for the circle Template:Math on the right, we have <math display=block>\left| \int_{C_\mathrm{R}} \frac{f(z)}{5-z} dz \right| \le 2 \pi \rho \frac{3.001^\frac34 \rho^\frac14}{1.999} \in \mathcal{O} \left( \rho^\frac54 \right) \to 0.</math>

Now using the Cauchy residue theorem, we have <math display=block>(-i + 1) I = -2\pi i \left( \operatorname{Res}_{z=5} \frac{f(z)}{5-z} + \operatorname{Res}_{z=\infty} \frac{f(z)}{5-z} \right).</math> where the minus sign is due to the clockwise direction around the residues. Using the branch of the logarithm from before, clearly <math display=block>\operatorname{Res}_{z=5} \frac{f(z)}{5-z} = - 5^\frac34 e^{\frac14 \log(-2)}.</math>

The pole is shown in blue in the diagram. The value simplifies to <math display=block>-5^\frac34 e^{\frac14(\log 2 + \pi i)} = -e^{\frac14 \pi i} 5^\frac34 2^\frac14.</math>

We use the following formula for the residue at infinity: <math display=block>\operatorname{Res}_{z=\infty} h(z) = \operatorname{Res}_{z=0} \left(- \frac{1}{z^2} h\left(\frac{1}{z}\right)\right).</math>

Substituting, we find <math display=block>\frac{1}{5-\frac{1}{z}} = -z \left(1 + 5z + 5^2 z^2 + 5^3 z^3 + \cdots\right)</math> and <math display=block>\left(\frac{1}{z^3}\left (3-\frac{1}{z} \right )\right)^\frac14 = \frac{1}{z} (3z-1)^\frac14 = \frac{1}{z}e^{\frac14 \pi i} (1-3z)^\frac14, </math> where we have used the fact that Template:Math for the second branch of the logarithm. Next we apply the binomial expansion, obtaining <math display=block>\frac{1}{z} e^{\frac14 \pi i} \left( 1 - {1/4 \choose 1} 3z + {1/4 \choose 2} 3^2 z^2 - {1/4 \choose 3} 3^3 z^3 + \cdots \right). </math>

The conclusion is that <math display=block>\operatorname{Res}_{z=\infty} \frac{f(z)}{5-z} = e^{\frac14 \pi i} \left (5 - \frac34 \right ) = e^{\frac14 \pi i}\frac{17}{4}.</math>

Finally, it follows that the value of Template:Mvar is <math display=block> I = 2 \pi i \frac{e^{\frac14 \pi i}}{-1+i} \left(\frac{17}{4} - 5^\frac34 2^\frac14 \right) = 2 \pi 2^{-\frac12} \left(\frac{17}{4} - 5^\frac34 2^\frac14 \right)</math> which yields <math display=block>I = \frac{\pi}{2\sqrt 2} \left(17 - 5^\frac34 2^\frac94 \right) = \frac{\pi}{2\sqrt 2} \left(17 - 40^\frac34 \right).</math>

Evaluation with residue theoremEdit

Using the residue theorem, we can evaluate closed contour integrals. The following are examples on evaluating contour integrals with the residue theorem.

Using the residue theorem, let us evaluate this contour integral. <math display=block>\oint_C \frac{e^z}{z^3}\,dz</math>

Recall that the residue theorem states <math display=block>\oint_{C} f(z) dz=2\pi i\cdot \sum\operatorname{Res}(f,a_k)</math>

where <math>\operatorname{Res}</math> is the residue of <math>f(z)</math>, and the <math>a_k</math> are the singularities of <math>f(z)</math> lying inside the contour <math>C</math> (with none of them lying directly on <math>C</math>).

<math>f(z)</math> has only one pole, <math>0</math>. From that, we determine that the residue of <math>f(z)</math> to be <math>\tfrac{1}{2}</math> <math display=block>\begin{align} \oint_C f(z) dz&=\oint_C \frac{e^z}{z^3}dz\\ &=2\pi i \cdot \operatorname{Res}_{z=0} f(z)\\ &=2\pi i\operatorname{Res}_{z=0} \frac{e^z}{z^3}\\ &=2\pi i \cdot \frac{1}{2}\\ &=\pi i \end{align}</math>

Thus, using the residue theorem, we can determine: <math display=block>\oint_C \frac{e^z}{z^3} dz = \pi i.</math>

Multivariable contour integralsEdit

To solve multivariable contour integrals (i.e. surface integrals, complex volume integrals, and higher order integrals), we must use the divergence theorem. For now, let <math>\nabla \cdot</math> be interchangeable with <math>\operatorname{div}</math>. These will both serve as the divergence of the vector field denoted as <math>\mathbf{F}</math>. This theorem states: <math display=block>\underbrace{\int \cdots \int_U}_n \operatorname{div}(\mathbf{F}) \, dV = \underbrace{ \oint \cdots \oint_{\partial U} }_{n-1} \mathbf{F} \cdot \mathbf{n} \, dS</math>

In addition, we also need to evaluate <math>\nabla\cdot \mathbf{F}</math> where <math>\nabla \cdot \mathbf{F}</math> is an alternate notation of <math>\operatorname{div} (\mathbf{F})</math>. The divergence of any dimension can be described as <math display=block>\begin{align} \operatorname{div}(\mathbf{F}) &=\nabla\cdot\mathbf{F}\\ &= \left(\frac{\partial}{\partial u}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}, \dots \right) \cdot (F_u,F_x,F_y,F_z,\dots)\\ &=\left(\frac{\partial F_u}{\partial u} + \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} + \cdots \right) \end{align}</math>

Example 1Edit

Let the vector field <math>\mathbf{F}=\sin(2x)\mathbf{e}_x+\sin(2y)\mathbf{e}_y+\sin(2z)\mathbf{e}_z</math> and be bounded by the following <math display=block>{0\leq x\leq 1} \quad {0\leq y\leq 3} \quad {-1\leq z\leq 4}</math>

The corresponding double contour integral would be set up as such: Template:Block indent

We now evaluate <math>\nabla\cdot\mathbf{F}</math>. Meanwhile, set up the corresponding triple integral: <math display=block>\begin{align} &=\iiint_V \left(\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\right) dV\\[6pt] &=\iiint_V \left(\frac{\partial \sin(2x)}{\partial x} + \frac{\partial \sin(2y)}{\partial y} + \frac{\partial \sin(2z)}{\partial z}\right) dV\\[6pt] &=\iiint_V 2 \left(\cos(2x) + \cos(2y) + \cos(2z)\right) dV \\[6pt] &=\int_{0}^{1}\int_{0}^{3}\int_{-1}^{4} 2(\cos(2x)+\cos(2y)+\cos(2z))\,dx\,dy\,dz \\[6pt] &=\int_{0}^{1}\int_{0}^{3}(10\cos(2y)+\sin(8)+\sin(2)+10\cos(z))\,dy\,dz\\[6pt] &=\int_{0}^{1}(30\cos(2z)+3\sin(2)+3\sin(8)+5\sin(6))\,dz\\[6pt] &=18\sin(2)+3\sin(8)+5\sin(6) \end{align}</math>

Example 2Edit

Let the vector field <math>\mathbf{F}=u^4\mathbf{e}_u+x^5\mathbf{e}_x+y^6\mathbf{e}_y+z^{-3}\mathbf{e}_z</math>, and remark that there are 4 parameters in this case. Let this vector field be bounded by the following: <math display=block>{0\leq x\leq 1} \quad {-10\leq y\leq 2\pi} \quad {4\leq z\leq 5} \quad {-1\leq u\leq 3}</math>

To evaluate this, we must utilize the divergence theorem as stated before, and we must evaluate <math>\nabla\cdot\mathbf{F}</math>. Let <math>dV = dx \, dy \, dz \, du</math>

Template:Block indent

<math display=block>\begin{align} &=\iiiint_V \left(\frac{\partial F_u}{\partial u} + \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\right)\,dV\\[6pt] &=\iiiint_V \left(\frac{\partial u^4}{\partial u} + \frac{\partial x^5}{\partial x} + \frac{\partial y^6}{\partial y} + \frac{\partial z^{-3}}{\partial z}\right)\,dV\\[6pt] &=\iiiint_V {{\frac{4 u^3 z^4 + 5 x^4 z^4 + 5 y^4 z^4 - 3}{z^4}}}\,dV \\[6pt] &= \iiiint_V {{\frac{4 u^3 z^4 + 5 x^4 z^4 + 5 y^4 z^4 - 3}{z^4}}}\,dV \\[6pt] &=\int_{0}^{1}\int_{-10}^{2\pi}\int_{4}^{5}\int_{-1}^{3} \frac{4 u^3 z^4 + 5 x^4 z^4 + 5 y^4 z^4 - 3}{z^4}\,dV\\[6pt] &=\int_{0}^{1}\int_{-10}^{2\pi}\int_{4}^{5}\left(\frac{4(3u^4z^3+3y^6+91z^3+3)}{3z^3}\right)\,dy\,dz\,du\\[6pt] &=\int_{0}^{1}\int_{-10}^{2\pi}\left(4u^4+\frac{743440}{21}+\frac{4}{z^3}\right)\,dz\,du\\[6pt] &=\int_{0}^{1} \left(-\frac{1}{2\pi^2}+\frac{1486880\pi}{21}+8\pi u^4+40 u^4+\frac{371720021}{1050}\right)\,du\\[6pt] &=\frac{371728421}{1050}+\frac{14869136\pi^3-105}{210\pi^2}\\[6pt] &\approx{576468.77} \end{align}</math>

Thus, we can evaluate a contour integral with <math>n=4</math>. We can use the same method to evaluate contour integrals for any vector field with <math>n>4</math> as well.

Integral representationEdit

Template:Expand section An integral representation of a function is an expression of the function involving a contour integral. Various integral representations are known for many special functions. Integral representations can be important for theoretical reasons, e.g. giving analytic continuation or functional equations, or sometimes for numerical evaluations.

File:Hankel contour.png
Hankel's contour

For example, the original definition of the Riemann zeta function Template:Math via a Dirichlet series, <math display=block>\zeta(s) = \sum_{n=1}^\infty\frac{1}{n^s},</math>

is valid only for Template:Math. But <math display=block>\zeta(s) = - \frac{\Gamma(1 - s)}{2 \pi i} \int_H\frac{(-t)^{s-1}}{e^t - 1} dt ,</math>

where the integration is done over the Hankel contour Template:Mvar, is valid for all complex s not equal to 1.

See alsoEdit

ReferencesEdit

Template:Reflist

Further readingEdit

External linksEdit

Template:Integral

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