advanced mathmatics for engineers ch.1

8/11/2019 Advanced Mathmatics for Engineers Ch.1

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Chapter 1

INTRODUCTION

The first and second semesters of calculus cover differential and integral calculus offunctions with only one variable. However, in reality, students have to deal with functionswith two or more variables. That is exactly what the third semester calculus is all about: the

differential and integral calculus of functions with two or more variables, usually calledmultiple variable calculus. Why bother? Calculus of functions of a single variable only provides a blurry image of a picture. Yet, many real world phenomena can only be described by functions with several variables. Calculus of functions of multiple variables perfects the picture and makes it more complete. Therefore, we introduce a few critical key points forfunctions of several variables in this book. Further details can always be found in a textbookfor multivariable calculus.

1.1. FUNCTIONS OF SEVERAL VARIABLES

We start with a simple example, as a function of two variablesand , shown in Figure 1.1.1.

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02

4 -4

-2

02

4-1

0

1

y-axisx-axis

z -

a x

i s

Figure 1.1.1. Graph of the two-variable function .


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Aliakbar Montazer Haghighi, Jian-ao Lian and Dimitar P. Mishev2

Definition 1.1.1.

A function of several variables is a function with domain being a subset of and range being a subset of , where the positive integer is the number of independent variables.

It is clear that explicit functions with two variables can in general be visualized or plotted, as the function shown in Figure 1.1.1.

Example 1.1.1.

Find the domain and range, and sketch the graph of the function of two variables.

SolutionThe domain of can be determined from the inequality , i.e., the

unit disk in : . If we let , the range is clearly

, simply because . Henceforward, the graph of is the top halfof the unit sphere in space, or . See Figure

1.1.2.

Definition 1.1.2.

A level set of a real-valued function of variables withrespect to an appropriate real value , called at level for short, is

.

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00.5

1

-1-0.5

00.5

1

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0

0.2

0.4

0.6

0.8

1

x-axisy-axis

z - a

x i s

Figure 1.1.2. Graph of the two-variable function , the unit hemisphere.


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Introduction 3

It is also called a contour of . A contour plot is a plot displaying level sets for severaldifferent levels. A level set of a function with two variables and at level is called alevel curve or a contour curve , i.e., . Level curves of the function in Figure 1.1.1at level is a collection of infinite circles

.

A level set of a function with three variables , , and at level is called a level surface or a contour surface , i.e., . The level surface of atlevel , which is nonnegative, is the sphere .

Functions with three or more variables cannot be visualized anymore. As a simpleexample, the function is a function with three variables. Although

cannot be visualized in the four-dimensional space, we can study the function byinvestigating its contour plots, i.e., for different level values on .

Definition 1.1.3.

A parametric equation or parametric form of a real-valued function or an equation is amethod of expressing the function or the equation by using parameters.

It is known that the equation of an ellipse in standard position, i.e., ,has its parametric form , , with the parameter . Similarly, astandard axis-aligned ellipsoid has a parametric form

(1.1.1)

where , which is used for azimuthal angle, or longitude (denoted by ), and, which is used for polar angle, or zenith angle, or colatitude (latitude ).

Equations in (1.1.1) are the usual spherical coordinates (radial, azimuthal, polar),a natural description of positions on a sphere.

A surface can sometimes also be given in terms of two parameters, or in its parametricform, like the function in Figure 1.1.1. Figure 1.1.3 is the plot of a typical seashell surface[cf., e.g., Davis & Sigmon 2005], given by a parametric form

(1.1.2)


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where and .

-4

0

4

-4

0

4

-9

-6

-3

0

Figure 1.1.3. Graph of a seashell surface with its parametric form given by (1.1.2).

Definition 1.1.4.

For , a -neighborhood (or -ball ) of a point in is the set of points that areless units away from , i.e., if

, the -neighborhood of is the set

.

This of course coincides with the usual -neighborhood of a point , when ,namely, the length open interval . When , the -neighborhood of a point is the open circle with diameter :

. Naturally, when , the -neighborhoodof a point is the following open sphere with diameter :

.

Definition 1.1.5.

Let be a function on with variables , and . Then the limit of is when approaches , denoted by


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Introduction 5

,

if for any , there is a such that for all points in the-neighborhood of , except possibly itself.

By , it means the point goes to by arbitrary orall possible ways. This indeed implies that

requires to be somehow well-defined in the -neighborhood of .

Example 1.1.2.

Find: (a) and (b) .

Solution

Let . It is clear that the domain of is all points in but the

origin, i.e., . Although the function is a rational

function, it is undefined at the origin. So, to find the limit of at , it is not simply

substituting both and by 0. Observe that

, which gives different

values when changes. In other words, if along two lines and ,the function goes to and , respectively. Therefore, does

not exist. Similarly, to find the limit in (b), we introduce . Then, for all

,

so that by using the fact that . Hence,

.

Definition 1.1.6.

A convenient notation called big O notation , describes the limiting behavior of a singlevariable real-valued function in terms of a simpler function when the argument of the functionapproaches a particular value or infinity. The first common big O notation

,


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meaning there are a sufficiently large number and a positive number such that

.

The second big O notation is

,

meaning there exist positive numbers and such that

.

The big O notation is very useful for determining the growth rate of functions. Forexample, the two limits in Example 1.1.3 suggest that the orders or speed when both thenumerator and denominator approaching 0 when play significant rule in

determining the existence of a limit. If we let , both the numerator anddenominator in (a) as but the numerator in (b) as and its

denominator as .

Definition 1.1.7.

Let be a function of variables with domain . The graph of is theset of all points such that for all

.

The graph of a function with two variables and and domain is the set of all points in

.

For example, Figure 1.1.1 shows the graph of .

1.2. PARTIAL DERIVATIVES , G RADIENT , AND DIVERGENCE

To extend the derivative concept of functions with a single variable to functions withseveral variables, we must use a new name partial derivative since all independent variableshave equal positions. Without loss of generality, let us concentrate on , a function withonly two variables and .


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Introduction 7

-4

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0

2

4 -4

-2

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y-axisx-axis

(a)

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0

2

4 -4

-2

0

2

4-10

0

10

y-axisx-axis

(b)

-4

-2

0

2

4 -4

-2

0

2

4-50

0

50

y-axisx-axis

(c)

Figure 1.1.4. (Continuted)


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-4

-2

0

2

4 -4

-2

0

2

4-20

0

20

y-axisx-axis

(d)

Figure 1.1.4. Graph of the partial derivatives , , , and of the two variable function

shown in Figure 1.1.1.

Definition 1.2.1.

The partial derivatives of with respect to and , denoted by and , or

and , respectively, are defined by

(1.2.1)

and. (1.2.2)

Clearly, the partial derivative of with respect to is simply the usual or regularderivative of a function with one variable , with in being considered as aconstant. The higher order part ial derivatives can be similarly defined, e.g.,

(1.2.3)

(1.2.4)

Figure 1.1.4 gives the plots of the four partial derivatives , , and for the

function plotted in Figure 1.1.1.


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Introduction 9

Example 1.2.1.

Find , , , , and , for .

Solution

For , we view as a constant in . So,

Similarly,

Next, to get , we again view all in the expression of as constants to get:

Analogously,

and

For high order partial derivatives, does the order with respect to different variablesmatter? In other words, is for a function in general?


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Theorem 1.2.1 ( Clairauts 1 Theorem or Schwarzs 2 Theorem )

If a function of variables has continuous second order partialderivatives at a point , then

, (1.2.5)

for . The continuity of is necessary for (1.2.5) to be true.

Example 1.2.2.

For find and .

SolutionFirst of all, is continuous at due to the fact that

Secondly, by using (1.2.1) and (1.2.2),

,

so that

and

1 Alexia Clairaut , May 3, 1713 May 17, 1765, was a prominent French mathematician, astronomer, geophysicist,and intellectual, and was born in Paris, France. http://en.wikipedia.org/wiki/Alexis_Clairaut.

2 Karl Hermann Amandus Schwarz , January 25, 1843 November 30, 1921, was a German mathematician, knownfor his work in complex analysis. He was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). There are alot of mathematical phrases being named after Schwarz s name, such as Cauchy-Schwarz inequality, Schwarzlemma, Schwarz triangle, just named a few. http://en.wikipedia.org/wiki/Hermann_Schwarz.


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Introduction 11

Thirdly, by using the definition for partial derivatives,

,

and

,

i.e., and . It then follows from (1.2.3) that

and

Hence, both and are discontinuous at , due to the fact

does not exist.

Definition 1.2.2.

The total differential or exact differential of a function of variables ,denoted by , is defined by

So the total differential of a function of two variables and is


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In a typical introductory differential equation course, there is a section called exactequations of first order ordinary differential equations (ODE). For example, the exact ODE

has solutions , where is an arbitrary constant, due to. In fact, the expression is a total

differential if and only if , provided that , , , , , and are allcontinuous. This is indeed a consequence of Theorem 1.2.1.

Example 1.2.3.

Solve the initial-value problem

.

SolutionFirst, rewrite the ODE into with

.

It is easy to verify that , i.e., the new ODE, namely,, is exact. To find a function so that and ,

we can start off by either integrating with respect to or integrating with respect to ,and then apply or . In details,

for an arbitrary function of . Then becomes

,

which leads to . Hence, the general solution for the first order ODE is

Definition 1.2.3.

The gradient or gradient vector is a (row) vector operator denoted by ,


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Introduction 13

where , , , are the unit vectors along the positive directions of axes , , ,. It is often applied to a real function of variables, e.g., , namely,

. (1.2.6)

For a real function of three variables, e.g., ,

. (1.2.7)

The notion of gradient has rich geometric interpretations. It can be used to define thetangent line and normal line of a surface at a point.

Definition 1.2.4.

The tangent plane of a surface at a point is

Definition 1.2.5.

The Laplace operator or Laplacian, denoted by , is defined from by

. (1.2.8)

In particular, for a function of three variables , , and ,

(1.2.9)

It was named after a French mathematician Pierre-Simon Laplace 3. The Laplacesequation is a partial differential equation (PDE) defined by

3 Pierre-Simon, marquis de Laplace, March 23, 1749 March 5, 1827, was a French mathematician andastronomer, and was born in Beaumont-en-Auge, Normandy. His work was pivotal to the development ofmathematical astronomy and statistics. The Bayesian interpretation of probability was mainly developed byLaplace. He formulated Laplace s equation, and pioneered the Laplace transform which appears in many


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(1.2.10)

All solutions of the Laplaces equations are called harmonic functions . Some simpleexamples of harmonic functions include , , , , and

. If the right-hand side is changed to a given function , then it is called Poissonsequation ,

(1.2.11)

It is named after another French mathematician Simon-Denis Poisson 4. Both theLaplaces equation and Poissons equation are the simplest elliptic partial differentialequations.

Example 1.2.4.

Given , find and .

SolutionFirst of all, all the partial derivatives needed are

,

,

,

.

Hence, it follows from (1.2.6) and (1.2.8) that

Definition 1.2.6.

The divergence of a vector field is an

operator defined by

branches of mathematical physics. The Laplace operator, widely used in applied mathematics, is also namedafter him. (http://en.wikipedia.org/wiki/Pierre-Simon_Laplace)

4 Simon-Denis Poisson, June 21, 1781 April 25, 1840, was a French mathematician, geometer, and physicist, andwas born in Pithiviers, Loiret. (http://en.wikipedia.org/wiki/Simeon_Poisson)


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Introduction 15

(1.2.12)

It is a scalar function and measures the magnit ude of a vector fields source or sink at a point. That is why it is sometimes also called flux density . A vector field isdivergence free (or divergenceless , or solenoidal ), if .

Definition 1.2.7.

The curl of a vector field is a matrix operator applied to a vector-valued function , defined by using the gradient operator ,

(1.2.13)

It measures the rotation of a vector field. So it is also sometimes called circulationdensity . The direction of the curl is determined by the right-hand rule. A vectorfield is irrotational or conservative if its curl is zero.

Example 1.2.5.

Given

,

find and .

SolutionLet , , and be the three components of . Then all the first order partial

derivatives of , and are


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Hence, it follows from (1.2.12) and (1.2.13) that

,

and

Definition 1.2.8.

The directional derivative of a function of variables at a pointalong a direction , denoted by or

, is the rate at which changes at in the direction , defined by the limit

.

So for a 2D function , its directional derivative at a point along a vector, denoted simply by , is

,

and for a 3D function , its directional derivative at a point along a vector, denoted simply by , is

.

The directional derivatives can be simply calculated by using the gradient, as indicated inthe following.


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Introduction 17

Theorem 1.2.2.

If all first order partial derivatives of at a point exist, then

. (1.2.14)

Example 1.2.6.

Find the derivative of the function at alongthe direction .

SolutionBy applying (1.2.14),

1.3. F UNCTIONS OF A C OMPLEX VARIABLE

Functions of a complex variable are complex-valued functions with an independentvariable being also complex-valued.

Definition 1.3.1.

A function of a complex variable , say, , defined on a set is a rule that assigns to everyin a complex number , called the function value of at , written as . The setis called the domain of and the set of all values of is called the range of .

Write , with and being the real and imaginary parts of . Sincedepends on , with and being the real and imaginary parts of , it is clear that

both and are real-valued functions of two real variables and . Hence, we may alsowrite

Example 1.3.1.

Let . Find the values of and at .


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SolutionBy simply replacing by , we have

so that

Hence, at ,

However, if we are not interested in the actual expressions of both and ,and at can also be easily obtained directly and quickly from at :

The complex natural exponential function is defined in terms of real-valued functions by

(1.3.1)

Clearly, if , meaning it is a natural extension of the real-valued naturalexponential function. When we have the Euler formula

(1.3.2)

It is named after the Swiss mathematician Leonhard Paul Euler 5. By using the usual polarcoordinates and for and , i.e.,

(1.3.3)

a complex number has its polar form

(1.3.4)

5 Leonhard Paul Euler, April 15, 1707 September 18, 1783, was a pioneering Swiss mathematician and physicist,and was born in Basel. Euler made important discoveries in calculus and graph theory. He introduced themathematical terminology such as function and mathematical analysis. A statement attributed to Pierr e-Simon Laplace expresses Euler s influence on mathematics: Read Euler, read Euler, he is the master of usall. (http://en.wikipedia.org/wiki/Leonhard_Euler)


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Introduction 19

Table 1.3.1. Some Elementary Complex-Valued Functions

Functions Period

Complexnaturalexponentialfunction

Complex

trigonometricfunctions

Complex

hyperbolicfunctions

Complexnaturallogarithmfunction

n/a

Complexgeneralexponentialfunction

, is a complex-valued constant n/a

Complexgeneral

powerfunction

, is a complex-valued constant,


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where is the absolute value or modulus of , denoted by , is the argument of , denoted by , namely:

(1.3.5)

(1.3.6)

and the Euler formula (1.3.2) was used in the last equality of (1.3.4). Obviously, the complexnatural exponential function is periodic with period . To define , we have to rewrite

in (1.3.6) more precisely, i.e.,

(1.3.7)

where represents the principle value of . So by defining as theinverse of the complex natural exponential function , we arrive at

(1.3.8)

where is the principle value of . The complex trigonometric functions and thecomplex hyperbolic functions can be defined naturally through

(1.3.9)

In summary, these and some other complex elementary functions are included in Table1.3.1.

1.4. P OWER SERIES AND THEIR C ONVERGENT BEHAVIOR

Complex power series are a natural extension of the real power series in calculus.Although all the properties and convergent tests of complex power series are completelysimilar to those of the real power series, complex power series have their own distinct

properties, e.g., they play a pivotal role for analytic functions in complex analysis. More precisely, complex power functions are analytic functions in complex analysis, and viseversa. For convenience, we will not separate real from complex power series, and simplyomitting the word complex in the sequel. It is easy to distinguish from the c ontext thatwhether it is real or complex. For examples, series may mean complex series, and powerseries may mean complex power series.

A series is the sum of a sequence, which can be finite or infinite. A finite series is a sumof finite sequences and an infinite series is the sum of an infinite sequence. Simple examplesof series are

(1.4.1)


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Introduction 21

and

(1.4.2)

where the former is a finite series of the arithmetic sequence

,

while the latter is an infinite series of the geometric sequence

By using the summation notations, these two series can be simply written as

(1.4.3)

respectively.

Definition 1.4.1.

A power series is an infinite series of the sum of all nonnegative integer powers of, namely,

(1.4.4)

where is a fixed constant or the center of the power series, , are constants orthe coefficients of the power series, and is a variable. The partial sum , denoted by

, is the sum of the first terms of the power series:

(1.4.5)

The remainder of the power series after the term , denoted by, is also an infinite series


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(1.4.6)

Definition 1.4.2.

A power series converges at , if its partial sum in (1.4.5) converges at as asequence, i.e.,

(1.4.7)

or, the sum of the series is , which is simply denoted by

Otherwise, it diverges , meaning the partial sum sequence diverges. For a convergentseries, its remainder , as a series, converges to 0. In fact, . A

power series always converges at its center (converges to ); and, in general, a powerseries has radius of convergence , if it converges for all in the -neighborhood of :

In the complex plane, denotes an open circular disk centered at andwith radius . The actual value of the radius of convergence of a power series can becalculated from either of the two formulations

(1.4.8)

provided the limit exists. The second formulation in (1.4.8) is also called Cauchy 6-Hadamard 7 formula.

6 Augustin-Louis Cauchy , August 21, 1789 May 23, 1857, was a French mathematician and was born in Paris,France. As a profound mathematician and prolific writer, Cauchy was an early pioneer of analysis, particularlycomplex analysis. His mathematical research spread out to basically the entire range of mathematics. Hestarted the project of formulating and proving the theorems of calculus in a rigorous manner and was thus anearly pioneer of analysis. He also gave several important theorems in complex analysis and initiated the studyof permutation groups. (http://en.wikipedia.org/wiki/Augustin_Louis_Cauchy)


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Introduction 23

Example 1.4.1.

Show that the geometric series converges for allsatisfying .

Proof

Due to the identity for any positive integer , it is

obvious that the geometric series converges to when (it diverges when

). It is also easy to see from (1.4.8) that the radius of convergence due to thefact that the coefficients . Hence,

(1.4.9)

Replacing by does not have any effect on the radius. So it is also true that

(1.4.10)

Example 1.4.2.

Find the power series expression for .

SolutionThe most trivial but important power series in mathematics is the one for the naturalexponential function, namely,

(1.4.11)

which has radius of convergence as since

In other words, (1.4.11) is an identity for all complex number .

7 Jacques Salomon Hadamard , December 8, 1865 October 17, 1963, was a French mathematician who bestknown for his proof of the prime number theorem in 1896. (http://en.wikipedia.org/wiki/Jacques_Hadamard)


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Example 1.4.3.

Find the radius of convergence for .

Solution

To find the radius of convergence for a series such as , it is helpful if we

apply the formulation to a generic geometric series. More precisely, the series can be viewedas

with . Hence, it converges when or , i.e.,

(1.4.12)

1.5. R EAL -V ALUED T AYLOR SERIES AND M ACLAURIN SERIES

To be easily moving to the complex-valued Taylor series and Maclaurin series, let us firstrecall real-valued Taylor series and Maclaurin series in calculus.

Definition 1.5.1.

The Taylor series of a real-valued function at is defined as the power series

(1.5.1)

provided that its order derivative exists at for all . Taylor series wasnamed in honor of the English mathematician Brook Taylor 8.

8 Brook Taylor , August 18, 1685 November 30, 1731, was an English mathematician and was born at Edmonton(or in then Middlesex). His name is attached to Taylors Theorem and the Taylor series. He was born atEdmonton (at that time in Middlesex), entered St John s College, Cambridge, as a fellow-commoner in 1701,


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Introduction 25

Definition 1.5.2.

The Maclaurin series of a real-valued function is the Taylor series of at , i.e.,

(1.5.2)

The Maclaurin series was named in honor of the Scottish mathematician ColinMaclaurin 9.

Again, for the geometric series , it is clear from

and that for all . Inother words, The geometric series is the Maclaurin series of the function

.

1.6. P OWER SERIES R EPRESENTATION OF ANALYTIC FUNCTIONS

1.6.1. Derivative and Analytic Functions

Definition 1.6.1.

A complex function is analytic at a point provided there exists such that itsderivative at exists, denoted by , i.e.,

(1.6.1)

exists for all in a -neighborhood of , i.e., , .There are real analytic functions and complex analytic functions. For simplicity, it is

common to use analytic functions exclusively for complex analytic functions. Simple

and took degrees of LL.B. and LL.D. respectively in 1709 and 1714. (http://en.wikipedia.org/wiki/Brook_Taylor)

9 Colin Maclaurin , February, 1698 June 14, 1746, was a Scottish mathematician, and was born in Kilmodan,Argyll. He entered the University of Glasgow at age eleven, not unusual at the time; but graduating MA by

successfully defending a thesis on the Power of Gravity at age 14. After graduation he remained at Glasgow tostudy divinity for a period, and in 1717, aged nineteen, after a competition which lasted for ten days, he waselected professor of mathematics at Marischal College in the University of Aberdeen. He held the record as theworlds youngest professor until Mar ch 2008. (http://en.wikipedia.org/wiki/Colin_Maclaurin) Side remark:Alia Sabur was three days short of her 19th birthday in February when she was hired to become a professor inthe Department of Advanced Technology Fusion at Konkuk University, Seoul. She is the youngest college

professor in history and broke Maclaurins almost 300 -year-old record. Alia Sabur was from New York andgraduated from Stony Brook University at age 14. She continued her education at Drexel University, whereshe earned an M.S. and a Ph.D. in materials science and engineering.


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examples of analytic functions include all complex-valued polynomial, exponential, andtrigonometric functions. Two typical and classical examples of non-analytic functions are theabsolute value function and the complex conjugate function

, where .

Definition 1.6.2.

A complex function is analytic (or holomorphic or regular ) on if it is analytic ateach point on ; it is entire if is analytic at each point of the complex plane.

Example 1.6.1.

Show by definition that the function for is nowhereanalytic.

ProofBy using the definition of in (1.6.1),

which is if and 1 if , i.e., does not exist at any point .

Example 1.6.2.

Show by definition that the function for is not analytic at any

point .

ProofAgain, by using (1.6.1),

.

Rationalize the numerator of the expression to get


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Introduction 27

When , it is , whose limit does not exist when . When , the second

term goes to when , but the first term does not converge when , also

due to does not exist. Therefore, the function is nowhere analytic.

Example 1.6.3.

Find the derivative of at .

SolutionAgain, it follows from (1.6.1) that

Another way of finding is to use the quotient rule, i.e.,

Back to definition (1.6.1), it can also be written as

(1.6.2)

where , , and . Then, when, and , , (1.6.2) leads to

, . (1.6.3)


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Theorem 1.6.1 ( Cauchy-Riemann 10 E quations and Analytic F unctions )

Let be a complex function, with and being continuouslydifferentiable on an open set . Then it is analytic on if and only if and satisfy theCauchy-Riemann equations

, (1.6.4)

. (1.6.5)

The necessity follows from (1.6.3), The proof of sufficiency is omitted since it is beyondthe scope of the book. We remark here that the Cauchy-Riemann equations in (1.6.3) has thefollowing equivalent forms

(1.6.6)

Moreover, if both and have continuous second order partial derivatives, we implyfrom (1.6.4) (1.6.5) that both and are harmonic functions, meaning they satisfy the 2DLaplaces equation in (1.2.10). Two harmonic functions and are called a harmonicconjugate pair if they satisfy the Cauchy-Riemann equations (1.6.4) (1.6.5).

It is now straightforward that both the functions and are notanalytic since their differentiable real and imaginary parts do not satisfy (1.6.4) (1.6.5).

Example 1.6.4.

Let , , which is a harmonic function. Findits harmonic conjugate such that the function is analytic.

SolutionIt follows from (1.6.4) (1.6.5) that

,

Hence, up to a constant, such a function is given by

10 Georg Friedrich Bernard Riemann , September 17, 1826 July 20, 1866, was a German mathematician and was

born in Breselenz, German. One of his most significant contributions is the Riemann hypothesis, a conjectureabout thedistribution of zeros of the Rieman zeta-function : All non-trivial zeros ofthe Riemann zeta-function have real part .


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Introduction 29

.

1.6.2. Line Integral in the Complex Plane

A definite integral of functions with one real variable in calculus can normally becalculated by applying the first fundamental theorem of calculus, if one of its antiderivativescan be explicitly given. Definite integrals for complex-valued functions, to be named complexline integrals, can also be similarly introduced.

Definition 1.6.3.

A simple curve is a curve that does not intersect or touch itself. A simple closed curve is aclosed simple curve. It is also sometimes called a contour . A domain is simply connected (also called 1-connected), if every simple closed curve within can be shrunk continuously

within to a point in (meaning without leaving ). A domain is multiply simplyconnected with holes (also called -connected), if it is formed by removing holes from a1-connected domain.

Definition 1.6.4.

A complex line integral is a definite integral of a complex-valued integrand withrespect to a complex integral variable over a given oriented curve in the complex plane,denoted by

(1.6.7)

if is closed. The curve is also called the path of integration .If is continuous and the curve is piecewise continuous, then the complex line

integral in (1.6.7) exists. If is in its parametric representation ,, the exact definition of a complex line integral is completely analogous to that of the

Riemann integral in calculus, namely: partition into subintervals according to the parameter ; form a Riemann sum; and take the limit when . We omit all details here.Furthermore, it is not hard to understand that the two integrals of over the two oppositeorientations of only differ up to a sign. Most properties for definite integrals of onevariable still carry over.


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Definition 1.6.5.

The second line integral in (1.6.7) is also called a contour integral .

If is analytic on a simply connected domain , then for any two points and in,

if , no matter how the curve is formed from to . This fact will be further

proved by the Cauchy Integral Theorem in 1.6.3.

Theorem 1.6.2.

Let be a harmonic function in a simply connected domain . Then all itsharmonic conjugates are given explicitly by

, (1.6.8)

where the integral is along any simple curve in from a point to , and isarbitrary.

Example 1.6.5.

Find the harmonic conjugates of , where is a constant.

SolutionBy applying (1.6.8) directly,


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Introduction 31

1.6.3. Cauchys Integral Theorem for Simply Connected Domains

The first look at the line integral (1.6.7) tells that it depends on not only the endpoints of but also the choice of the path of itself. The following theorem tells that a line integral is

independent of the path if is analytic in a simply connected domain and is a piecewise continuous curve in .

Theorem 1.6.3 ( Cauchys Integral Theorem )

Let be an analytic function in a simply connected domain and an arbitrarysimple closed positively oriented contour that lies in . Then

(1.6.9)

Example 1.6.6.

Evaluate for any contour .

SolutionSince is entire, the line integral is 0 on any contour . Indeed, if is entire,

on any contour . Similarly, , and

for .

Example 1.6.7.

Evaluate , where is the counterclockwise unit circle.

SolutionIt follows from the parametric form of the unit circle:

that

Notice that the Cauchys integral theorem cannot be applied on Example 1.6.7. Thecontour (unit circle) can be included within the annulus for anyand . The integrand is analytic on this annulus but it is not simply connected.


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Example 1.6.8.

Evaluate , where is the counterclockwise unit circle and is a constant that is

different from .

SolutionSimilar to Example 1.6.7,

which is 0 for any integer different from . If we take the limit when , its value isas it was shown in Example 1.6.7.Example 1.6.8 tells that the analytic property for in Theorem 1.6.3 is sufficient but

not necessary (for a contour integral to vanish).

Example 1.6.9.

Evaluate , where is the counterclockwise unit circle.

SolutionAgain, similar to Example 1.6.7,

We knew in Section 1.6.1 that was nowhere analytic. So result of Example 1.6.9 does

not conflict with Theorem 1.6.3. However, when is the counterclockwise

unit circle, since on and the constant 1 is entire.

1.6.4. Cauc hys Integral Theorem for Multiply Connected Domains

Cauchys integral T heorem 1.6.3 for simply connected domains can be generalized tomultiply connected domains.

Theorem 1.6.4. ( Cauchys Integral Theorem for Multiply Connected Domains )

Let be an analytic function in a -connected domain , and let an arbitrarysimple closed positively oriented contour that lies in . Then


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Introduction 33

(1.6.10)

where is the outer boundary (simple closed) curve with counterclockwise orientation and, , , are the inner simple closed curves of the holes inside with clockwise

orientation.

1.6.5. Cauchys Integral Formula

A good starting point to understand analytic functions is the following, i.e., an integralcan represent any analytic function.

Theorem 1.6.5. ( Cauchys Integral Formula )

Let be an analytic function in the simply connected domain and a simple

closed positively oriented contour that lies in . If is a point that lies in interior to , then

(1.6.11)

Here, the contour integral is taken on , who is positively oriented, meaning counter-clockwise .

Example 1.6.10.

Evaluate , where is the unit circle with positive orientation.

SolutionIt follows from that the only zero of

that lies in the interior of is . So by letting , theintegral

It also follows from this example that

.


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Example 1.6.11.

Evaluate , where is the circle with positive orientation.

SolutionIntroduce . Then, due to the fact that

it follows from Theorem 1.6.5 that

1.6.6. Cauchys Integral Formula for Derivatives

For derivatives of analytic functions, see the following.

Theorem 1.6.6. ( Cauchys In tegral F ormula f or D eri vatives )

Let be an analytic function in the simply connected domain , and a simpleclosed positively oriented contour that lies in . If is a point that lies in interior to , thenfor any nonnegative integer ,

(1.6.12)

Example 1.6.12.

Evaluate , where is the unit circle with counterclockwise orientation.


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Introduction 35

SolutionSince is entire and the unit circle encloses , by a direct application of (1.6.12)

in Theorem 1.6.6,

Example 1.6.13.

Evaluate , where consists of the circle

(counterclockwise) and the circle (clockwise).

Solution

Denote by the annulus enclosed by and let (see Table

1.3.1). Then is analytic in and the point is enclosed in the interior of . It followsfrom Theorem 1.6.6 that

1.6.7. Taylor and Maclaurin Series of Complex-Valued Functions

The real-valued Taylor series can now be extended to the complex-valued.

Definition 1.6.6.

The Taylor series of a function centered at is defined as the power series

, (1.6.13)

Definition 1.6.7.

Similarly, the Maclaurin series is when :


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. (1.6.14)

Table 1.6.1 includes a few common but important Maclaurin series.

Table 1.6.1. A Few Important Maclaurin Series

Maclaurin series of Radius ofConvergent

ConvergentInterval for

real-valued

Convergentregion forcomplex-valued

1

1

1

Theorem 1.6.7. ( Taylor Theorem )

If is analytic on and , then the Taylor seriesof converges to for all , i.e.,

(1.6.15)

Furthermore, the convergence is uniform on for any .

From Theorem 1.6.5, an analytic function can also be defined by a function that is locallygiven by a convergent power series.


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Introduction 37

Definition 1.6.8.

An analytic function on an open set is a complex-valued infinitely differentiable

function such that its Taylor series at any pointconverges to for all in a neighborhood of , i.e.,

(1.6.16)

where is a small positive number depending on .

1.6.8. Taylor Polynomials and their Applications

Definition 1.6.9.

An degree Taylor polynomial of a real-valued function is the degree polynomial truncated from or a partial sum of the Taylor series of , as the firstterms, namely,

. (1.6.17)

Recall from Calculus I that if a function is differentiable at a point , the linearapproximation to in a neighborhood of was indeedin (1.6.17) when . In other words, the linear approximation from Calculus

I is now expended to any degree polynomial approximation by using the degreeTaylor polynomial .


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E XERCISES

1.1. F UNCTIONS OF S EVERAL V ARIABLES

For Problems 1.1.1 10, find and sketch the domain of the functions of two variables.

1.1.1. .

1.1.2. .

1.1.3. .

1.1.4. .

1.1.5. .

1.1.6. .

1.1.7. .

1.1.8. .

1.1.9. .

1.1.10 . .

For Problems 1.1.11 21, find the limits if they exist.

1.1.11 . .

1.1.12 . .

1.1.13. .


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Introduction 39

1.1.14. .

1.1.15. .

1.1.16. .

1.1.17. .

1.1.18. .

1.1.19. .

1.1.20. .

1.1.21. .

1.2. P ARTIAL D ERIVATIVES , G RADIENT , AND D IVERGENCE

For 1.2.1 5, find the indicated partial derivatives.

1.2.1. Let . Find the values of and at the point .

1.2.2. For , find and .

1.2.3. Find and when .

1.2.4. Find and if , , and satisfy .

1.2.5. Find , , where , , , and satisfy .


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For 1.2.6 10, find the first order partial derivatives.

1.2.6. .

1.2.7. .

1.2.8. .

1.2.9. .

1.2.10 . .

For 1.2.11 13, find the second order partial derivatives.

1.2.11 . .

1.2.12 . .

1.2.13 . .

1.2.14 . Find the gradient of the function at the point .

1.2.15 . Find the derivative of the function at the pointin the direction .

1.2.16 . The volume of a cone is given by the equation . Use the total

differential to estimate the change in volume if the height increases from 10 to 10.1 cm andthe radius decreases from 12 to 11.95 cm.

1.2.17. Find the linearization of at the point .

1.2.18. Three resistors are connected in parallel as shown:


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Introduction 41

The resistance . Find the percent change in if , , and increase

by 1 , 0 and , and .

For 1.2.19 22, find the derivative of the function at the given point in the givendirection .

1.2.19. , , .

1.2.20. , , .

1.2.21. , , and .

1.2.22. , , and .

1.3. F UNCTIONS OF A C OMPLEX V ARIABLE

For 1.3.1 4, find and .

1.3.1. .

1.3.2. .

1.3.3. .

1.3.4. .

For 1.3.5 8, find the limits.

1.3.5. .

1.3.6. .

1.3.7. .

1.3.8. .


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For 1.3.9 17 , evaluate.

1.3.9. .

1.3.10. .

1.3.11. .

1.3.12. .

1.3.13 . .

1.3.14 . .

1.3.15 . .

1.3.16 . .

1.3.17 . .

For 1.3.18 20, solve the equations for .

1.3.18. .

1.3.19. .

1.3.20. .

1.4. P OWER S ERIES AND THEIR C ONVERGENT B EHAVIOR

1.4.1. Find the sums of both the finite series and infinite series in (1.4.1) and (1.4.2).

1.4.2. Derive by using (1.4.9) when the generic geometric series convergesand what it converges to. In addition, when does it diverge?

1.4.3. The sum of reciprocals of natural numbers is called the harmonic series , namely,

. Show that the harmonic series is divergent.


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Introduction 43

1.4.4. By using the partial faction, show that the series

converges for any fixed real number . What does it converge to?

For 1.4.5 18, determine whether the given series is convergent or divergent. If it is, findits sum.

1.4.5. .

1.4.6. .

1.4.7. .

1.4.8. .

1.4.9. .

1.4.10. .

1.4.11. .

1.4.12. .

1.4.13. .

1.4.14. .

1.4.15. .


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1.4.16. .

1.4.17. .

1.4.18. .

For 1.4.19 24, find the radius of convergence for the power series.

1.4.19. .

1.4.20. .

1.4.21. .

1.4.22. .

1.4.23. .

1.4.24. .

1.5. R EAL -V ALUED T AYLOR S ERIES AND M ACLAURIN S ERIES

For 1.5.1 7 , find the Maclaurin series expansion of the functions.

1.5.1. .

1.5.2. .

1.5.3. .


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Introduction 45

1.5.4. .

1.5.5. .

1.5.6. .

1.5.7. .

1.5.8. Find the first five terms in the Taylor series about for .

1.5.9. Find the interval of convergence for the series in Problem 1.5.8.

1.5.10. Use partial fractions and the result from Problem 1.5.8 to find the first five terms

in the Taylor series about for .

1.5.11. Let be the function defined by . Write the first four terms and

the general term of the Taylor series expansion of about .1.5.12 . Use the result from Problem 1.5.11 to find the first four terms and the general

term of the series expansion about for , .

1.5.13. Find the Taylor series expansion about for .

1.6. P OWER S ERIES R EPRESENTATION OF A NALYTIC F UNCTIONS

For 1.6.1 4, find the Derivatives of the functions.

1.6.1. .

1.6.2. .

1.6.3. .

1.6.4. .

For 1.6.5 8, verify the Cauchy-Riemann equations for the functions.

1.6.5. .


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1.6.6. .

1.6.7. .

1.6.8. .

For 1.6.9 13, evaluate the given integral, where is the circle with positive orientation.

1.6.9. , .

1.6.10. , .

1.6.11. , .

1.6.12. , .

1.6.13. , .

For 1.6.14 17 , find the Maclaurin series expansion of the functions.

1.6.14. .

1.6.15. .

1.6.16. .

1.6.17. .



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