Chapter 14

Complex Variables

A deeper understanding of functional analysis, even principles involving real functions of real variables, can be attained if the functions and variables are extended into the complex plane. Figure 14.1 shows schematically how a functional relationship image can be represented by a mapping of the z-plane, with image, into the w-plane, with image.

image

Figure 14.1 Mapping of the functional relation image. If the function is analytic, then the mapping is conformal, with orthogonality of grid lines preserved.

14.1 Analytic Functions

Let

image (14.1)

be a complex-valued function constructed from two real functions image and image. Under what conditions can image be considered a legitimate function of a single complex variable image, allowing us to write image? A simple example would be

image (14.2)

so that

image (14.3)

Evidently

image (14.4)

This function can be represented in the complex plane as shown in Figure 14.2. A counterexample, which is not a legitimate function of a complex variable, would be

image (14.5)

since the complex conjugate image is not considered a function of z. To derive a general condition for image, express x and y in terms of z and image using

image (14.6)

An arbitrary function in Eq. (14.1) can thus be reexpressed in the functional form image. The condition that image, with no dependence on image, implies that

image (14.7)

We can write

image (14.8)

Using (14.6), this reduces to

image (14.9)

Since the real and imaginary parts must individually equal zero, we obtain the Cauchy-Riemann equations:

image (14.10)

These conditions on the real and imaginary parts of a function image must be fulfilled in order for w to a function of the complex variable z. If, in addition, u and image have continuous partial derivatives with respect to x and y in some region, then image in that region is an analytic function of z. In complex analysis, the term holomorphic function is often used to distinguish it from a real analytic function.

image

Figure 14.2 Contours of image in the complex plane: image

A complex variable z can alternatively be expressed in polar form

image (14.11)

where image is referred to as the modulus and image, the phase or argument. Correspondingly, the function image would be written

image (14.12)

The Cauchy-Riemann equations in polar form are then given by

image (14.13)

Consider the function

image (14.14)

Both image and image and their x- and y-derivatives are well behaved everywhere in the image-plane except at the point image, where they become discontinuous and, in fact, infinite. In mathematical jargon, the function and its derivatives do not exist at that point. We say therefore that image is an analytic function in the entire complex plane except at the point image. A value of z at which a function is not analytic is called a singular point or singularity.

Taking the x-derivative of the first Cauchy-Riemann equation and the y-derivative of the second, we have

image (14.15)

Since the mixed second derivatives of image are equal,

image (14.16)

Analogously, we find

image (14.17)

Therefore both the real and imaginary parts of an analytic function are solution of the two-dimensional Laplace equation, known as harmonic functions. This can be verified for image and image, given in the above example of the analytic function image.

14.2 Derivative of an Analytic Function

The derivative of a complex function is given by the obvious transcription of the definition used for real functions:

image (14.18)

In the definition of a real derivative, such as image or image, there is only one way for image or image to approach zero. For image in the complex plane, there are however an infinite number of ways to approach image. For an analytic function, all of them should give the same result for the derivative.

Let us consider two alternative ways to achieve the limit image: (1) along the x-axis with image or (2) along the y-axis with image. With image, we can write

image (14.19)

with

image (14.20)

The limits for image in the alternative processes are then given by

image (14.21)

and

image (14.22)

Equating the real and imaginary parts of (14.21) and (14.22), we again arrive at the Cauchy-Riemann Eqs. (14.10).

All the familiar formulas for derivatives remain valid for complex variables, for example, image, and so forth.

14.3 Contour Integrals

The integral of a complex function image has the form of a line integral (see Section 11.7) over a specified path or contour C between two points image and image in the complex plane. It is defined as the analogous limit of a Riemann sum:

image (14.23)

where the points image lie on a continuous path C between image and image. In the most general case, the value of the integral depends on the path C. For the case of an analytic function in a simply connected region, we will show that the contour integral is independent of path, being determined entirely by the endpoints image and image.

14.4 Cauchy’s Theorem

This is the central result in the theory of complex variables. It states that the line integral of an analytic function around an arbitrary closed path in a simple-connected region vanishes:

image (14.24)

The path of integration is understood to be traversed in the counterclockwise sense. An “informal” proof can be based on the identification of image with an exact differential expression (see Section 11.6):

image (14.25)

It is seen that Euler’s reciprocity relation (11.48)

image (14.26)

is equivalent to the Cauchy-Riemann Eqs. (14.10). Cauchy’s theorem is then a simple transcription of the result (11.73) for the line integral around a closed path. The region in play must be simply connected, with no singularities. Equation (14.24) is sometimes referred to as the Cauchy-Goursat theorem. Goursat proved it under somewhat less restrictive conditions, showing that image need not be a continuous function.

14.5 Cauchy’s Integral Formula

The most important applications of Cauchy’s theorem involve functions with singular points. Consider the integral

image

around the closed path C shown in Figure 14.3. Let image be an analytic function in the entire region. Then image is also analytic except at the point image. The contour C can be shrunken to a small circle image surrounding image, as shown in the figure. The infinitesimally narrow channel connecting C to image is traversed in both directions, thus canceling its contribution to the integral around the composite contour. By Cauchy’s theorem

image (14.27)

The minus sign appears because the integration is clockwise around the circle image. We find therefore

image (14.28)

assuming that image is a sufficiently small circle that image is nearly constant within, well approximated as image. It is convenient now to switch to a polar representation of the complex variable, with

image (14.29)

We find then

image (14.30)

The result is Cauchy’s integral theorem:

image (14.31)

A remarkable implication of this formula is a sort of holographic principle. If the values of an analytic function image are known on the boundary of a region, then the value of the function can be determined at every point image inside that region.

image

Figure 14.3 Contours for derivation of Cauchy’s integral formula: image

Cauchy’s integral formula can be differentiated with respect to image any number of times to give

image (14.32)

and, more generally,

image (14.33)

This shows, incidentally, that derivatives of all orders exist for an analytic function.

14.6 Taylor Series

Taylor’s theorem can be derived from the Cauchy integral theorem. Let us first rewrite (14.31) as

image (14.34)

where image is now the variable of integration along the contour C and z, any point in the interior of the contour. Let us develop a power-series expansion of image around the point image, also within the contour. Applying the binomial theorem, we can write

image (14.35)

Note that image so that image and the series converges. Substituting the summation into (14.34), we obtain

image (14.36)

Therefore, using Cauchy’s integral theorem (14.33),

image (14.37)

This shows that a function analytic in a region can be expanded in a Taylor series about a point image within that region. The series (14.37) will converge to image within a certain radius of convergence, a circle of radius image, equal to the distance to image, the singular point closest to image.

We can now understand the puzzling behavior of the series

image (14.38)

which we encountered in Eq. (7.37). The complex function image has a singularity at image. Thus an expansion about image will be valid only within a circle of radius of 1 around the origin. This means that a Taylor series about image will be valid only for image. On the real axis this corresponds to image and means that both the series image and image will converge only under this condition. The function image could, however, be expanded about image, giving a larger radius of convergence image. Along the real axis, we find

image (14.39)

which converges for image.

The process of shifting the domain of a Taylor series is known as analytic continuation. Figure 14.4 shows the circles of convergence for image expanded about image and about image. Successive applications of analytic continuation can cover the entire complex plane, exclusive of singular points (with some limitations for multivalued functions).

image

Figure 14.4 Analytic continuation of series expansion for image with singularity at image. Expansion about image converges inside small circle, while expansion about image converges inside large circle.

14.7 Laurent Expansions

Taylor series are valid expansions of image about points image (sometimes called regular points) within the region where the function is analytic. It is also possible to expand a function about singular points. Figure 14.5 outlines an annular (shaped like a lock washer) region around a singularity image of a function image, but avoiding other singularities at image and image. The function is integrated around the contour including image in a counterclockwise sense, image in a clockwise sense, and the connecting cut in canceling directions. Denoting the complex variable on the contour by image, we can apply Cauchy’s theorem to obtain

image (14.40)

where z is any point within the annular region. On the contour image we have image so that image, validating the convergent expansion (14.35):

image (14.41)

On the contour image, however, image so that image and we have instead

image (14.42)

where we have inverted the fractions in the last summation and shifted the dummy index. Substituting the last two expansions into (14.40), we obtain

image (14.43)

This is a summation containing both positive and negative powers of image:

image (14.44)

known as a Laurent series. The coefficients are given by

image (14.45)

where C is any counterclockwise contour within the annular region encircling the point image. The result can also be combined into a single summation

image (14.46)

with image now understood to be defined for both positive and negative n.

image

Figure 14.5 Contour for derivation of Laurent expansion of image about singular point image. The singularities at image and image are avoided.

When image for all n, the Laurent expansion reduces to an ordinary Taylor series. A function with some negative power of image in its Laurent expansion has, of necessity, a singularity at image. If the lowest negative power is image (with image for image), then image is said to have a pole of order N at image. If image, so that image is the lowest-power contribution, then image is called a simple pole. For example, image has a simple pole at image and a pole of order 2 at image. If the Laurent series does not terminate, the function is said to have an essential singularity. For example, the exponential of a reciprocal,

image (14.47)

has an essential singularity at image. The poles in a Laurent expansion are instances of isolated singularities, to be distinguished from continuous arrays of singularities which can also occur.

14.8 Calculus of Residues

In a Laurent expansion for image within the region enclosed by C, the coefficient image (or image) of the term image is given by

image (14.48)

This is called the residue of image and plays a very significant role in complex analysis. If a function contains several singular points within the contour C, the contour can be shrunken to a series of small circles around the singularities image, as shown in Figure 14.6. The residue theorem states that the value of the contour integral is given by

image (14.49)

If a function image, as represented by a Laurent series (14.44) or (14.46), is integrated term by term, the respective contributions are given by

image (14.50)

Only the contribution from image will survive—hence the designation “residue.”

image

Figure 14.6 The contour for the integral image can be shrunken to enclose just the singular points of image. This is applied in derivation of the theorem of residues.

The residue of image at a simple pole image is easy to find:

image (14.51)

At a pole of order N, the residue is a bit more complicated:

image (14.52)

The calculus of residues can be applied to the evaluation of certain types of real integrals. Consider first a trigonometric integral of the form

image (14.53)

With a change of variables to image, this can be transformed into a contour integral around the unit circle, as shown in Figure 14.7. Note that

image (14.54)

so that image can be expressed as image. Also image. Therefore the integral becomes

image (14.55)

and can be evaluated by finding all the residues of image inside the unit circle:

image (14.56)

As an example, consider the integral

image (14.57)

This is equal to the contour integral

image (14.58)

where

image (14.59)

The pole at image lies outside the unit circle when image. Thus we need include only the residue of the integrand at image:

image (14.60)

Finally, therefore,

image (14.61)

image

Figure 14.7 Evaluation of trigonometric integral: image

An infinite integral of the form

image (14.62)

can also be evaluated by the calculus of residues provided that the complex function image is analytic in the upper half plane with a finite number of poles. It is also necessary for image to approach zero more rapidly than image as image in the upper half plane. Consider, for example,

image (14.63)

The contour integral over a semicircular sector shown in Figure 14.8 has the value

image (14.64)

On the semicircular arc image, we can write image so that

image (14.65)

Thus, as image, the contribution from the semicircle vanishes while the limits of the x-integral extend to image. The function image has simple poles at image. Only the pole at image is in the upper half plane, with image, therefore

image (14.66)

Problem 14.8.1

Evaluate the following integrals:

image

image

Figure 14.8 Evaluation of image by contour integration in the complex plane. Only singularities in the upper half plane contribute.

14.9 Multivalued Functions

Thus far we have considered single-valued functions, which are uniquely specified by an independent variable z. The simplest counterexample is the square root image which is a two-valued function. Even in the real domain, image can equal either image. When the complex function image is expressed in polar form

image (14.67)

it is seen that the full range of image requires that image vary from 0 to image (not just image). This means that the complex z-plane must be traversed twice in order to attain all possible values of image. The extended domain of z can be represented as a Riemann surface—constructed by duplication of the complex plane, as shown in Figure 14.9. The Riemann surface corresponds to the full domain of a complex variable z. For purposes of visualization, the surface is divided into connected Riemann sheets, each of which is a conventional complex plane. Thus the Riemann surface for image consists of two Riemann sheets connected along a branch cut, which is conveniently chosen as the negative real axis. A Riemann sheet represents a single branch of a multivalued function. For example, the first Riemann sheet of the square-root function produces values image in the range image, while the second sheet is generated by image. A point contained in every Riemann sheet, image in the case of the square-root function, is called a branch point. The trajectory of the branch cut beginning at the branch point is determined by convenience or convention. Thus the branch cut for image could have been chosen as any path from image to image.

image

Figure 14.9 Representations of Riemann surface for image. The dashed segments of the loops lie on the second Riemann sheet.

The Riemann surface for the cube root image comprises three Riemann sheets, corresponding to three branches of the function. Analogously, any integer or rational power of z will have a finite number of branches. However, an irrational power such as image will not be periodic in any integer multiple of image and will hence require an infinite number of Riemann sheets. The same is true of the complex logarithmic function

image (14.68)

and of the inverse of any periodic function, including image, etc. In such cases, the Riemann surface can be imagined as an infinite helical (or spiral) ramp, as shown in Figure 14.10.

image

Figure 14.10 Schematic representation of several sheets of the Riemann surface needed to cover the domain of a multivalued function such as image, or image.

Branch cuts can be exploited in the evaluation of certain integrals, for example

image

with image. Consider the corresponding complex integral around the contour shown in Figure 14.11. A small and a large circle of radii image and image, respectively, are joined by a branch cut along the positive real axis. We can write

image (14.69)

Along the upper edge of the branch cut we take image. Along the lower edge, however, the phase of z has increased by image, so that, in noninteger powers, image. In the limit as image and image, the contributions from both circular contours approach zero. The only singular point within the contour C is at image, with residue image. Therefore

image (14.70)

and finally

image (14.71)

Problem 14.9.1

Evaluate the integral

image

image

Figure 14.11 Contour used to evaluate the integral image

14.10 Integral Representations for Special Functions

Some very elegant representations of special functions are possible with use of contour integrals in the complex plane.

Recall Rodrigues’ formula for Legendre polynomials (13.78):

image (14.72)

Applying Cauchy’s integral formula (14.33) to image, we obtain

image (14.73)

This leads to Schlaefli’s integral representation for Legendre polynomials:

image (14.74)

where the path of integration is some contour enclosing the point image.

A contour-integral representation for Hermite polynomials can be deduced from the generating function (13.124), rewritten as

image (14.75)

Dividing by image and taking a contour integral around the origin:

image (14.76)

By virtue of (14.50), only the image term in the summation survives integration, leading to the result:

image (14.77)

An analogous procedure works for Laguerre polynomials. From the generating function (13.150)

image (14.78)

we deduce

image (14.79)

Bessel functions of integer order can be found from the generating function (13.42):

image (14.80)

This suggests the integral representation:

image (14.81)

For Bessel functions of noninteger order image, the same integral pertains except that the contour must be deformed as shown in Figure 14.12, to take account of the multivalued factor image. The contour surrounds the branch cut along the negative real axis, such that it lies entirely within a single Riemann sheet.

image

Figure 14.12 Contour for representation (14.81) of Bessel function image of noninteger order.

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