{"id":989,"date":"2018-11-22T11:27:59","date_gmt":"2018-11-22T11:27:59","guid":{"rendered":"http:\/\/msp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=989"},"modified":"2019-04-30T12:04:13","modified_gmt":"2019-04-30T12:04:13","slug":"analytic-functions","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp01\/chapter\/analytic-functions\/","title":{"rendered":"Analytic functions"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\nTABLE OF CONTENTS<\/strong>\r\n\r\n1.\u00a0 Polynomials and rational functions\r\n\r\n2.\u00a0 Power series\r\n

2.1 Cauchy\u2019s n<\/em>th root test<\/p>\r\n

2.2 Power series and analytic functions<\/p>\r\n3. Common functions\r\n

3.1 The exponential function<\/p>\r\n

3.2 The trigonometric functions<\/p>\r\n

3.2.1 The hyperbolic functions<\/p>\r\n

3.2.2 Zeros of sin z<\/em> and cos z<\/em><\/p>\r\n

3.3 Periodicity of exp z<\/em><\/p>\r\n

3.4 The logarithmic function<\/p>\r\n

3.4.1 Power series for log(1+z<\/em>)<\/p>\r\n

3.5 The function<\/p>\r\n \r\n\r\n \r\n

LEARNING OBJECTIVES<\/strong><\/p>\r\n

\"\"<\/p>\r\n \r\n

A n a l y t i c\u00a0 f u n c t i o n s<\/strong><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n1. Polynomials and rational functions<\/span>\r\n

We now wish to study some special functions with which we are familiar from study of functions of a real variable. We would like to extend their definition to the entire complex plane and study their analyticity properties.<\/p>\r\n

The power function zn<\/sup><\/em>, where n<\/em> is a positive integer, is one of the simplest analytic function. It is an entire function. That means it has no singularity in the finite complex plane. The derivative of the function is<\/p>\r\n\"\"\r\n\r\n\"\"\r\n

Since positive integer powers are entire functions, it follows that a polynomial of order n<\/em> is also an entire function. That is, the function<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\nA function of the form\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n2. Power series<\/strong><\/span>\r\n

An infinite series whose terms consist of positive integer powers of z<\/em> (in ascending order) is called a power series<\/em>:<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n2.1 Cauchy\u2019s n<\/em>th<\/sup> root test<\/span>\r\n

Cauchy\u2019s root test is a very useful test for the convergence of an infinite series. Consider the series<\/p>\r\n\"\"\r\n\r\n<\/div>\r\nLet<\/span>\r\n\r\n\"\"\r\n

where \u201clim sup\u201d stands for the superior or upper limit.\u00a0 The root test states that<\/span><\/p>\r\nif <\/span>M<\/em> < 1, then the series converges absolutely,<\/span>\r\n\r\nif <\/span>M<\/em> > 1, then the series diverges,<\/span>\r\n\r\nif <\/span>M<\/em> = 1, and the limit approaches strictly from above, then the series diverges.<\/span>\r\n\r\nIn the cases excluded, the test is inconclusive \u2013 the series may diverge, converge absolutely or may converge conditionally.<\/span>\r\n\r\n\"\"\r\n

There are three cases to be considered: (i) <\/span>R<\/em> = 0, (ii) <\/span>R<\/em> finite and (iii) <\/span>R<\/em> infinite. The first case, <\/span>R<\/em> = 0, is a trivial case \u2013 the series converges only for <\/span>z<\/em> = 0. In the third case, <\/span>R<\/em> infinite, the series converges for all <\/span>z<\/em>.<\/span><\/p>\r\n\r\n

\r\n\r\n \r\n\r\n2.2 Power series and analytic functions<\/span>\r\n

We have the theorem that if a power series has a nonzero radius of convergence then its sum is an analytic<\/em> function regular within its circle of convergence<\/em>.<\/p>\r\n \r\n\r\nProof<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nThe radius of convergence of each of these series is <\/span>R<\/em>.<\/span>\r\n\r\n\"\"\r\n\r\n \r\n\r\n3. Common functions<\/span><\/strong><\/span>\r\n\r\n<\/div>\r\n
\r\n

3.1 The exponential function<\/span><\/p>\r\n

We will now introduce some of the elementary functions with which we are already familiar. However, now we will introduce them as functions of a complex variable via the power series and study their properties; some of these properties may be different and others extensions of those of functions of a real variable. We begin with the well known exponential function.<\/p>\r\nThe exponential function<\/em> exp(z<\/em>) is defined by the power series\r\n\r\n\"\"\r\n\r\n\"\"\r\n

Hence the exponential function is an entire function, that is, it has no singularities in the entire finite complex plane. When z<\/em> is a real number x<\/em>, this function is the same as the function of the elementary algebra. The exponential function exp is also often written as ez<\/sup><\/em>. This is because like the function of real variable, the exponential function of the complex variable also satisfies the relation<\/p>\r\n\"\"\r\n\r\n \r\n\r\nProof<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n3.2 The trigonometric functions<\/span>\r\n

The elementary definition of trigonometric functions is via geometry where they are defined as ratios of various sides of right-angled triangles. It can then be shown that if x<\/em> is the measure of an angle in radians, then<\/p>\r\n\"\"\r\n

for all values of the real variable x<\/em>. We now define<\/em> the trigonometric functions \u201csine\u201d and \u201ccosine\u201d of a complex variable z<\/em> by<\/p>\r\n\"\"\r\n

From the ratio test it follows immediately that both these series have an infinite radius of convergence. Hence these represent functions that are analytic and have no singularities in the entire finite complex plane. We can obtain the derivatives of these functions by term by term differentiation of the terms of the series; the result is<\/p>\r\n\"\"\r\n\r\nThe other usual trigonometric functions can be defined by\r\n\r\n\"\"\r\n\r\nFrom these definitions we can find the derivatives of these functions:\r\n\r\n\"\"\r\n

The functions tan z<\/em> and sec z<\/em> are analytic in any finite domain in which cos z<\/em> does not vanish; cot z<\/em> and cosec z<\/em> in any finite domain in which sin z<\/em> does not vanish.<\/p>\r\n

We can immediately realize, on comparing series for the exponential and the trigonometric functions, that<\/p>\r\n\"\"\r\n

On using the addition formula for the exponential function we can easily prove the following well known relations<\/p>\r\n\"\"\r\n\r\nAll these, in fact most, relations that are true for real <\/span>x<\/em> are also true for complex <\/span>z<\/em>.<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n3.2.1 The hyperbolic functions<\/span>\r\n\r\nFor real x<\/em> the hyperbolic functions are defined by the relations\r\n\r\n\"\"\r\n\r\nWe define the hyperbolic functions for all real and complex variables by the same relations:\r\n\r\n\"\"\r\n\r\nThe other hyperbolic functions are also defined in analogy with the case of real variables:\r\n\r\n\"\"\r\n

These are all analytic functions. Whereas sinh z<\/em> and cosh z<\/em> are entire functions having no singularities in any finite domain, tanh z<\/em> and sech z<\/em> have singularities at the points where cosh z<\/em> vanishes and cot z<\/em> and cosech z<\/em> have singularities where sinh z<\/em> vanishes.<\/p>\r\nThe following relations follow directly from the definitions of the trigonometric and hyperbolic functions:\r\n\r\n\"\"\r\n

Through these equations we can relate the properties of the hyperbolic functions to those of the trigonometric functions.<\/p>\r\n \r\n\r\n3.2.2. Zeros of sin z<\/em> and cos z<\/em><\/span>\r\n

We will now use the above relations (16) and (17) and the addition formulae for sines and cosines, equations (12) and (13), to find the zeros of the sine and cosine functions for complex z<\/em>.<\/p>\r\nLet z = x + iy<\/em>. Then\r\n\r\n\"\"\r\n

The arguments of all the functions are now real variables; thus we have separated the function sin z<\/em> into its real and imaginary parts. Therefore, it follows that sin z<\/em> will be zero if, and only if,<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n3.3 Periodicity of exp <\/span>z<\/em><\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n

When written in the exponent form this gives<\/span><\/p>\r\n\"\"\r\n

Equating real and imaginary parts of this equation, we have<\/span><\/p>\r\n\"\"\r\n

On squaring an adding these two equations we have<\/span><\/p>\r\n\"\"\r\n

Thus we have proved that the exponential is a periodic function with a fundamental period <\/span>2\u03c0i<\/em>.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n3.4 The logarithmic function<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

Since the argument of a complex number has infinitely many values, differing from each other by multiples of 2<\/span>\u03c0<\/em>, a complex number has infinite number of logarithms. We write all these together as<\/span><\/p>\r\n\"\"\r\n

For each <\/span>z<\/em>, Log <\/span>z<\/em> is an infinite valued function. Each value of Log <\/span>z<\/em> obtained by choosing a specific value of the argument is called a <\/span>branch<\/em> of the logarithm. The most important branch is the one corresponding to the principal value of the argument and is called the <\/span>principal value of the logarithm<\/em> of <\/span>z<\/em>. This principal value is denoted by log <\/span>z<\/em>. Thus<\/span><\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n3.4.1 Power series for log(1+z<\/em>)<\/span>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nprovided the series for f(<\/span>z<\/em>) converges.\u00a0 Now consider the function<\/span>\r\n\r\n\"\"\r\n

This function is analytic and regular for |<\/span>z<\/em>| < 1 and has zero derivative. Hence it is a constant, independent of <\/span>z<\/em>. By putting <\/span>z<\/em> = 0 in the above equation, this constant is found to be zero. Hence<\/span><\/p>\r\n\"\"\r\n\r\nThe series converges for |<\/span>z<\/em>| < 1.<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n3.5 The function<\/span>\r\n\r\nIf m<\/em> is an integer then the function\r\n\r\n\"\"\r\n

Though Log z<\/em> is an infinite valued function, exp(m<\/em> Log z<\/em>) is single valued.\u00a0 If n<\/em> is any other integer, the function<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nThe other (n<\/em> \u2013 1) solutions are then\r\n\r\n\"\"\r\n

Each of these solutions can be reached from others by going around the branch point at z<\/em> = 0.<\/p>\r\n\"\"\r\n

The law of indices for real variables is valid for this definition as well:<\/p>\r\n\"\"\r\n\r\nAs we have already seen, log z<\/em> is discontinuous along the negative real axis. Its value changes by 2\u03c0i<\/em> in going from y<\/em>+<\/sup> to y<\/em>-<\/sup>. Hence\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

<\/div>\r\n \r\n\r\nSUMMARY<\/strong>\r\n