Chapter 7

Series and Integrals

When you press the SIN or LOG key on your scientific calculator, it almost instantly returns a numerical value. What is really happening is that the microprocessor inside the calculator is summing a series representation of that function, similar to the series we have already encountered for image, image, and image in Chapter 3 and 4. Power series (sums of powers of image) and other series of functions are very important mathematical tools, both for computation and for deriving theoretical results.

7.1 Some Elementary Series

An arithmetic progression is a sequence such as 1, 4, 7, 10, 13, 16. As shown in Section 1.2, the sum of an arithmetic progression with image terms is given by

image (7.1)

where image is the first term, image is the constant difference between terms, and image is the last term.

A geometric progression is a sequence which increases or decreases by a common factor image, for example, 1, 3, 9, 27, 81, … or 1, 1/2, 1/4, 1/8, 1/16, … The sum of a geometric progression is given by,

image (7.2)

When image, the sum can be carried to infinity to give

image (7.3)

We had already found from the binomial theorem that

image (7.4)

By applying the binomial theorem successively to image, then to each image you can show that

image (7.5)

Therefore image serves as a generating function for the Fibonacci numbers:

image (7.6)

An infinite geometric series inspired by one of Zeno’s paradoxes is

image (7.7)

Zeno’s paradox of motion claims that if you shoot an arrow, it can never reach its target. First it has to travel half way, then half way again—meaning 1/4 of the distance—then continue with an infinite number of steps, each taking it image closer. Since infinity is so large, you’ll never get there! What we now understand that Zeno possibly didn’t (some scholars believe that his argument was meant to be satirical) was that an infinite number of decreasing terms can add up to a finite quantity.

The integers 1 to to image add up to

image (7.8)

while the sum of the squares is given by

image (7.9)

and the sum of the cubes by

image (7.10)

7.2 Power Series

Almost all functions can be represented by power series of the form

image (7.11)

where image might be a factor such as image, image, or image. The case image and image provides the most straightforward class of power series. We are already familiar with the series for the exponential function:

image (7.12)

as well as the sine and cosine:

image (7.13)

image (7.14)

Recall also the binomial expansion:

image (7.15)

Useful results can be obtained when power series are differentiated or integrated term by term. This is a valid procedure under very general conditions.

Consider, for example, the binomial expansion

image (7.16)

Making use of the known integral

image (7.17)

we obtain a series for the arctangent

image (7.18)

With image, this gives a famous series for image

image (7.19)

usually attributed to Gregory and Leibniz.

A second example begins with another binomial expansion

image (7.20)

We can again evaluate the integral

image (7.21)

This gives a series representation for the natural logarithm:

image (7.22)

For image, this gives another famous series

image (7.23)

As a practical matter, the convergence of this series is excruciatingly slow. It takes about 1000 terms to get image correct to three significant figures, image.

It is relatively simple to multiply two power series. Note that

image (7.24)

Problem 7.2.1

Evaluate the first few terms of the power series for image and image.

Inversion of a power series means, in principle, solving for the expansion variable image as a function of the sum of the series. Consider the special case of a series in the form

image (7.25)

The constants image and image in the more general case can be absorbed into the variable image and the remaining constants. Then image can be expanded in powers of image in the form

image (7.26)

By a recursive procedure, carried out only as far as image, we obtain:

image (7.27)

Problem 7.2.2

Invert the power series for image to find the first few terms in the power series for image.

7.3 Convergence of Series

The partial sumimage of an infinite series is the sum of the first image terms:

image (7.28)

The series is convergent if

image (7.29)

where image is a finite quantity. A necessary condition for convergence, thus a preliminary test, is that

image (7.30)

Several tests for convergence that are usually covered in introductory calculus courses. The comparison test: if a series image is known to converge and image for all image, then the series image is also convergent. The ratio test: if image then the series converges. There are more sensitive ratio tests in the case that the limit approaches 1, but you will rarely need these outside of math courses. The most useful test for convergence is the integral test. This is based on turning things around using our original definition of an integral as the limit of a sum. The sum image can be approximated by an integral by turning the discrete variable image into a continuous variable image. If the integral

image (7.31)

is finite, then the original series converges.

A general result is that any decreasing alternating series, such as (7.23), converges. Alternating refers to the alternation of plus and minus signs. How about the analogous series with all plus signs?:

image (7.32)

After 1000 terms the sum equals 7.485. It might appear that, with sufficient patience, the series will eventually converge to a finite quantity. Not so, however! The series is divergent and sums to infinity. This can be seen by applying the integral test:

image (7.33)

A finite series of the form

image (7.34)

is called a harmonic series. Using the same approximation by an integral, we estimate that this sum is approximately equal to image. The difference between image and image was shown by Euler to approach a constant as image:

image (7.35)

where image (sometimes denoted image) is known as the Euler-Mascheroni constant. It comes up frequently in mathematics, for example, the integral

image (7.36)

An alternating series is said to be absolutely convergent if the corresponding sum of absolute values, image, is also convergent. This is not true for the alternating harmonic series (7.23). Such a series is said to be conditionally convergent. Conditionally convergent series must be treated with extreme caution. A theorem due to Riemann states that, by a suitable rearrangement of terms, a conditionally convergent series may be made to equal any desired value, or to diverge. Consider, for example, the series (7.20) for image when image:

image (7.37)

Different ways of grouping the terms of the series give different answers. Thus image, while image. But image actually equals 1/2. Be very careful!

7.4 Taylor Series

There is a systematic procedure for deriving power-series expansions for all well-behaved functions. Assuming that a function image can be represented in a power series about image, we write

image (7.38)

Clearly, when image,

image (7.39)

The first derivative of image is given by

image (7.40)

and, setting image, we obtain

image (7.41)

The second derivative is given by

image (7.42)


image (7.43)

With repeated differentiation, we find

image (7.44)


image (7.45)

We have therefore determined the coefficients image in terms of derivatives of image evaluated at image and the expansion (7.38) can be given more explicitly by

image (7.46)

If the expansion is carried out around image, rather than 0, the result generalizes to

image (7.47)

This result is known as Taylor’s theorem and the expansion is a Taylor series. The case image, given by Eq. (7.46), is sometimes called a Maclaurin series.

In order for a Taylor series around image to be valid it is necessary for all derivatives image to exist. The function is then said to be analytic at image. A function which is not analytic at one point can still be analytic at other points. For example, image is not analytic at image but is at image. The series (7.22) is equivalent to an expansion of image around image.

We can now systematically derive all the series we obtained previously by various other methods. For example, given image, we find

image (7.48)

so that (7.47) gives

image (7.49)

which is the binomial expansion. This result is seen to be valid even for noninteger values of image. In the latter case we should express (7.49) in terms of the gamma function as follows:

image (7.50)

The series for image is easy to derive because image for all image. Therefore, as we have already found

image (7.51)

The Taylor series for image is also straightforward since successive derivatives cycle among image, image, image, and image. Since image and image, the series expansion contains only odd powers of image with alternating signs:

image (7.52)

Analogously, the expansion for the cosine is given by

image (7.53)

Euler’s theorem image can then be deduced by comparing the series for these three functions.

Problem 7.4.1

Work out the Taylor series for the function image about image, up to the term in image. This is a little tricky since you have to evaluate limits as image to determine the derivatives image.

7.5 Bernoulli and Euler Numbers

The answer to Problem 7.4.1 is the series

image (7.54)

This series provides a generating function for the Bernoulli numbers, image, whereby

image (7.55)


image (7.56)

Bernoulli numbers find application in number theory, in numerical analysis, and in expansions of several functions related to image and image. A symbolic relation which can be used to evaluate Bernoulli numbers mimics the binomial expansion:

image (7.57)

where image is to be replaced by image.

We can obtain an expansion for image in the following steps involving the Bernoulli numbers:

image (7.58)


image (7.59)


image (7.60)

and, more explicitly,

image (7.61)

Problem 7.5.1

Find the expansions for image and for image.

Problem 7.5.2

From the trigonometric identity image, work out the expansions for image.

Somewhat analogous to the definition of Bernoulli numbers are the Euler numbersimage. These can be obtained from the generating function:

image (7.62)

The first few Euler numbers are

image (7.63)

with all equal to 0 for odd indices image. We find directly that

image (7.64)

and thus obtain the expansion

image (7.65)

Problem 7.5.3

Find the corresponding expansion for image.

The Euler-Maclaurin sum formula provides a powerful method for evaluating some difficult summations. It actually represents a more precise connection between Riemann sums and their corresponding integrals. It can be used to approximate finite sums and even infinite series using integrals with some additional terms involving Bernoulli numbers. It can be shown that

image (7.66)

Problem 7.5.4

Using the Euler-Maclaurin formula, evaluate the sum image.

7.6 L’Hôpital’s Rule

The value of a function is called an indeterminate form at some point image if its limit as image apparently approaches one of the forms image, or image. Two examples are the combinations image and image as image. (As we used to say in high school, such sick functions had to be sent to L’Hôspital to be cured.) To be specific, let us consider a case for which

image (7.67)

If image and image are both expressed in Taylor series about image,

image (7.68)

If image and image both equal 0 but image and image are finite, the limit in (7.67) is given by

image (7.69)

a result known as L’Hôpital’s rule. In the event that one or both first derivatives also vanishes, the lowest order nonvanishing derivatives in the numerator and denominator determine the limit.

To evaluate the limit of image, let image and image. In this case, image. But image, so image. Also, image, for all image. Therefore

image (7.70)

This is also consistent with the approximation that image for image. For the limit of image, let image, image. Now image and image. We find therefore that

image (7.71)

Problem 7.6.1

Evaluate the limit


7.7 Fourier Series

Taylor series expansions, as described above, provide a very general method for representing a large class of mathematical functions. For the special case of periodic functions, a powerful alternative method is expansion in an infinite sum of sines and cosines, known as a trigonometric series or Fourier series. A periodic function is one which repeats in value when its argument is increased by multiples of a constant image, called the period or wavelength. For example,

image (7.72)

as shown in Figure 7.1. For convenience, let us consider the case image. Sine and cosine are definitive examples of functions periodic in image since

image (7.73)

The functions image and image, with image=integer, are also periodic in image (as well as in image).


Figure 7.1 Periodic function image with period image.

A function image periodic in image can be expanded as follows:

image (7.74)

Writing the constant term as image is convenient, as will be seen shortly. The coefficients image can be determined by making use of the following definite integrals:

image (7.75)

image (7.76)

image (7.77)

These integrals are expressed compactly with use of the Kronecker delta, defined as follows:

image (7.78)

Two functions are said to be orthogonal if the definite integral of their product equals zero. (Analogously, two vectors image and image whose scalar product image equals zero are said to be orthogonal—meaning perpendicular, in that context.) The set of functions image is termed an orthogonal set in an interval of width image.

The nonvanishing integrals in (7.75) and (7.76) follow easily from the fact that image and image have average values of image over an integral number of wavelengths. Thus

image (7.79)

The orthogonality relations (7.75)(7.77) enable us to determine the Fourier expansion coefficients image and image. Consider the integral image, with image expanded using (7.74). By virtue of the orthogonality relations, only one term in the expansion survives integration:

image (7.80)

Solving for image we find

image (7.81)

Note that the case image correctly determines image, by virtue of the factor image in Eq. (7.74). Analogously, the coefficients image are given by

image (7.82)

In about half of textbooks, the limits of integration in Eqs. (7.81) and (7.82) are chosen as image rather than image. This is just another way to specify one period of the function and the same results are obtained in either case.

As an illustration, let us calculate the Fourier expansion for a square wave, defined as follows:

image (7.83)

A square-wave oscillator is often used to test the frequency response of an electronic circuit. The Fourier coefficients can be computed using (7.81) and (7.82). First we find

image (7.84)

Thus all the cosine contributions equal zero since image is symmetrical about image. The square wave is evidently a Fourier sine series, with only nonvanishing image coefficients. We find

image (7.85)


image (7.86)

The Fourier expansion for a square wave can thus be written

image (7.87)

For image, Eq. (7.87) reduces to (7.19), the Gregory-Leibniz series for image:

image (7.88)

A Fourier series carried through image terms is called a partial sumimage. Figure 7.2 shows the function image and the partial sums image, and image. Note how the higher partial sums overshoot near the points of discontinuity image. This is known as the Gibbs phenomenon. As image at these points.

Problem 7.7.1

Calculate the first several terms of the Fourier series representing a sawtooth signal, shown below.



Figure 7.2 Fourier series approximating square wave image. Partial sums image, and image are shown.

The general conditions for a periodic function image to be representable by a Fourier series are the following. The function must have a finite number of maxima and minima and a finite number of discontinuities between 0 and image. Also, image must be finite. If these conditions are fulfilled then the Fourier series (7.74) with coefficients given by (7.81) and (7.82) converges to image at points where the function is continuous. At discontinuities image, the Fourier series converges to the midpoint of the jump, image.

Recall that sines and cosines can be expressed in terms of complex exponential functions, according to Eqs. (4.60) and (4.61). Accordingly, a Fourier series can be expressed in a more compact form:

image (7.89)

where the coefficients image might be complex numbers. The orthogonality relations for complex exponentials are given by

image (7.90)

These determine the complex Fourier coefficients:

image (7.91)

If image is a real function, then image for all image.

For functions with a periodicity image different from image, the variable image can be replaced by image. The formulas for Fourier series are then modified as follows:

image (7.92)


image (7.93)


image (7.94)

For complex Fourier series,

image (7.95)


image (7.96)

Many of the applications of Fourier analysis involve the time-frequency domain. A time-dependent signal image can be expressed

image (7.97)

where the image are frequencies expressed in radians per second.

When a tuning fork is struck, it emits a tone which can be represented by a sinusoidal wave—one having the shape of a sine or cosine. For tuning musical instruments, a fork might be machined to produce a pure tone at 440 Hz, which corresponds to A above middle C. (Middle C would then have a frequency of 278.4375 Hz.) The graph in Figure 7.3 shows the variation of air pressure (or density) with time for a sound wave, as measured at a single point. image represents the deviation from the undisturbed atmospheric pressure image. The maximum variation of image above or below image is called the amplitudeimage of the wave. The time between successive maxima of the wave is called the periodimage. Since the argument of the sine or cosine varies between image and image in one period, the form of the wave could be the function

image (7.98)

Psi (image) is a very common symbol for wave amplitude. The frequencyimage, defined by

image (7.99)

gives the number of oscillations per second, conventionally expressed in hertz (Hz). An alternative measure of frequency is the number of radians per second, image. Since one cycle corresponds to image radians,

image (7.100)

The upper strip in Figure 7.3 shows the profile of the sound wave at a single instant of time. The pressure or density of the air also has a sinusoidal shape. At some given instant of time image the deviation of pressure from the undisturbed presssure image might be represented by

image (7.101)

where image is the wavelength of the sound, the distance between successive pressure maxima. Sound consists of longitudinal waves, in which the wave amplitude varies in the direction parallel to the wave’s motion. By contrast, electromagnetic waves, such as light, are transverse, with their electric and magnetic fields oscillating perpendicular to the direction of motion. The speed of light in vacuum, image m/s. The speed of sound in air is much slower, typically around 350 m/s (1100 ft/s or 770 miles/hr—known as Mach 1 for jet planes). As you know, thunder is the sound of lightning. You see the lightning essentially instantaneously but thunder takes about 5 s to travel 1 mile. You can calculate how far away a storm is by counting the number of seconds between the lightning and the thunder. A wave (light or sound) traveling at a speed image moves a distance of one wavelength image in the time of one period image. This implies the general relationship between frequency and wavelength

image (7.102)

valid for all types of wave phenomena. A trumpet playing the same sustained note produces a much richer sound than a tuning fork, as shown in Figure 7.4. Fourier analysis of a musical tone shows a superposition of the fundamental frequency image augmented by harmonics or overtones, which are integer multiples of image.


Figure 7.3 Sound wave of a single frequency produced by a tuning fork. The upper part of the figure shows an instantaneous view of a longitudinal sound wave traveling at speed image. Wavelength and period are related by image.


Figure 7.4 Fourier analysis of trumpet playing single note of frequency image Hz, one octave above middle C. Upper curve represents the signal in the time domain, lower curve, in the frequency domain.

7.8 Dirac Deltafunction

Recall that the Kronecker delta image, defined in Eq. (7.78), pertains to the discrete variables image and image. A useful application enables the reduction of a summation to a single term:

image (7.103)

The analog of the Kronecker delta for continuous variables is the Dirac deltafunctionimage, which has the defining property

image (7.104)

which includes the normalization condition

image (7.105)


image (7.106)

The approach to image is sufficiently tame, however, that the integral has a finite value.

A simple representation for the deltafunction is the limit of a normalized Gaussian as the standard deviation approaches zero:

image (7.107)

The Dirac deltafunction is shown pictorially in Figure 7.5. The deltafunction is the limit of a function which becomes larger and larger in an interval which becomes narrower and narrower. (Some university educators bemoaning increased specialization contend that graduate students are learning more and more about less and less until they eventually wind up knowing everything about nothing—the ultimate deltafunction!)


Figure 7.5 Dirac deltafunction with image as limit of normalized Gaussian: image.

Differentiation of a function at a finite discontinuity produces a deltafunction. Consider, for example, the Heaviside unit step function:

image (7.108)

Sometimes image (for image) is defined as image. The derivative of the Heaviside function image is clearly equal to zero when image. In addition

image (7.109)

We can thus identify

image (7.110)

The deltafunction can be generalized to multiple dimensions. In three dimensions, the defining relation for a deltafunction can be expressed

image (7.111)

For example, the limit of a continuous distribution of electrical charge image shrunken to a point charge image at image can be represented by

image (7.112)

The potential energy of interaction between two continuous charge distributions is given by

image (7.113)

If the distribution image is reduced to a point charge image at image, this reduces to

image (7.114)

If the analogous thing then happens to image, the formula reduces to the Coulomb potential energy between two point charges

image (7.115)

Problem 7.8.1

Show that the normalized Lorentzian distribution


approaches a deltafunction in the limit as image.

7.9 Fourier Integrals

Fourier series are ideal for periodic functions, sums over frequencies which are integral multiples of some image. For a more general class of functions which are not simply periodic, all possible frequency contributions must be considered. This can be accomplished by replacing a discrete Fourier series by a continuous integral. The coefficients image (or, equivalently, image and image) which represent the relative weight of each harmonic will turn into a Fourier transformimage, which measures the contribution of a frequency in a continuous range of image. In the limit as image, a complex Fourier series (7.95) generalizes to a Fourier integral. The discrete variable image can be replaced by a continuous variable image, such that

image (7.116)

with the substitution

image (7.117)

Correspondingly, Eq. (7.96) becomes

image (7.118)

where image is called the Fourier transform of image—alternatively written image, image, image, image or sometimes simply image. A Fourier-transform pair image and image can also be defined more symmetrically by writing:

image (7.119)

Fourier integrals in the time-frequency domain have the form

image (7.120)

Figure 7.4 is best described as a Fourier transform of a trumpet tone since the spectrum of frequencies consists of peaks of finite width.

Another representation of the deltafunction, useful in the manipulation of Fourier transforms, is defined by the limit:

image (7.121)

For image, the function equals image, which approaches image. For image the sine function oscillates with ever-increasing frequency as image. The positive and negative contributions cancel so that the function becomes essentially equivalent to 0 under the integral sign. Finally, since

image (7.122)

for finite values of k, Eq. (7.121) is suitably normalized to represent a deltafunction. The significance of this last representation is shown by the integral

image (7.123)

This shows that the Fourier transform of a complex monochromatic wave image is a deltafunction image.

An important result for Fourier transforms can be derived using (7.123). Using the symmetrical form for the Fourier integral (7.119), we can write

image (7.124)

being careful to use the dummy variable image in the second integral on the right. The integral over image on the right then gives

image (7.125)

Following this by integration over image we obtain

image (7.126)

a result known as Parseval’s theorem. A closely related result is Plancherel’s theorem:

image (7.127)

where image and image are the symmetric Fourier transforms of image and image, respectively.

The convolution of two functions is defined by

image (7.128)

The Fourier transform of a convolution integral can be found by substituting the symmetric Fourier transforms image and image and using the deltafunction formula (7.125). The result is the convolution theorem, which can be expressed very compactly as

image (7.129)

In an alternative form, image.

7.10 Generalized Fourier Expansions

For Bessel functions and other types of special functions to be introduced later, it is possible to construct orthonormal sets of basis functions image which satisfy the orthogonality and normalization conditions:

image (7.130)

with respect to integration over appropriate limits. (If the functions are real, complex conjugation is unnecessary.) An arbitrary function image in the same domain as the basis functions can be expanded in an expansion analogous to a Fourier series

image (7.131)

with the coefficients determined by

image (7.132)

If (7.132) is substituted into (7.131), with the appropriate use of a dummy variable, we obtain

image (7.133)

The last quantity in square brackets has the same effect as the deltafunction image. The relation known as closure

image (7.134)

is, in a sense, complementary to the orthonormality condition (7.130).

Generalized Fourier series find extensive application in mathematical physics, particularly quantum mechanics.

7.11 Asymptotic Series

In certain circumstances, a divergent series can be used to determine approximate values of a function as image. Consider, as an example, the complementary error function image, defined in Eq. (6.95):

image (7.135)

Noting that

image (7.136)

Equation (7.135) can be integrated by parts to give

image (7.137)

Integrating by parts again:

image (7.138)

Continuing the process, we obtain

image (7.139)

This is an instance of an asymptotic series, as indicated by the equivalence symbol image rather than an equal sign. The series in brackets is actually divergent. However, a finite number of terms gives an approximation to image for large values of image. The omitted terms, when expressed in their original form as an integral, as in Eqs. (7.137) or (7.138), approach zero as image.

Consider the general case of an asymptotic series

image (7.140)

with a partial sum

image (7.141)

The condition for image to be an asymptotic representation for image can be expressed

image (7.142)

A convergent series approaches image for a given image as image, where image is the number of terms in the partial sum image. By contrast, an asymptotic series approaches image as image for a given image.

The exponential integral is another function defined as a definite integral which cannot be evaluated in a simple closed form. In the usual notation

image (7.143)

By repeated integration by parts, the following asymptotic series for the exponential integral can be derived:

image (7.144)

Finally, we will derive an asymptotic expansion for the gamma function, applying a technique known as the method of steepest descents. Recall the integral definition:

image (7.145)

For large image the integrand image will be very sharply peaked around image. It is convenient to write

image (7.146)

A Taylor series expansion of image about image gives

image (7.147)

noting that image. The integral (7.145) can then be approximated as follows:

image (7.148)

For large image, we introduce a negligible error by extending the lower integration limit to image. The integral can then be done exactly to give Stirling’s formula

image (7.149)

or in terms of the factorial

image (7.150)

This is consistent with the well-known approximation for the natural logarithm:

image (7.151)

A more complete asymptotic expansion for the gamma function will have the form

image (7.152)

Making use of the recursion relation

image (7.153)

it can be shown that image and image.

You will notice that, as we go along, we are leaving more and more computational details for you to work out on your own. Hopefully, your mathematical facility is improving at a sufficient rate to keep everything understandable.

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