Looking for GATE Preparation Material? Join & Get here now!

** Gate 2013 Question Papers.. ** CEED 2013 Results.. ** Gate 2013 Question Papers With Solutions.. ** GATE 2013 CUT-OFFs.. ** GATE 2013 Results.. **

Next>>

Complex Functions and Linear Mappings

Complex Functions and Linear Mappings

Chapter 2Complex Functions

Overview

The last chapter developed a basic theory of complex numbers. For the next few chapters we turn our attention to functions of complex numbers. They are defined in a similar way to functions of real numbers that you studied in calculus; the only difference is that they operate on complex numbers rather than real numbers. This chapter focuses primarily on very basic functions, their representations, and properties associated with functions such as limits and continuity. You will learn some interesting applications as well as some exciting new ideas.

2.1 Functions and Linear Mappings

A complex-valued function f of the complex variable z is a rule that assigns to each complex number z in a set D one and only one complex number w.We writeand call w the image of z under f.A simple example of a complex-valued function is given by the formula.The set D is called the domain of f, and the set of all imagesis called the range of f.When the context is obvious, we omit the phrase complex-valued, and simply refer to a function f, or to a complex function f.

We can define the domain to be any set that makes sense for a given rule, so for,we could have the entire complex plane for the domain D, or we might artificially restrict the domain to some set such as.Determining the range for a function defined by a formula is not always easy, but we will see plenty of examples later on.In some contexts functions are referred to as mappings or transformations.

In Section 1.6, we used the term domain to indicate a connected open set.When speaking about the domain of a function, however, we mean only the set of points on which the function is defined.This distinction is worth noting, and context will make clear the use intended.

Just as z can be expressed by its real and imaginary parts,,we write,where u and v are the real and imaginary parts of w, respectively.Doing so gives us the representation

.

Because u and v depend on x and y, they can be considered to be real-valued functions of the real variables x and y; that is,

and.

Combining these ideas, we often write a complex function f in the form

.

Figure 2.1 illustrates the notion of a function (mapping) using these symbols.

[Graphics:Images/ComplexFunLinear_gr_12.gif]

Figure 2.1The mapping.

There are two methods for defining a complex function in Mathematica.

We now give several examples that illustrate how to express a complex function.

Example 2.1.Writein the for.

Solution.Using the binomial formula, we obtain

[Graphics:Images/ComplexFunLinear_gr_25.gif]

so that.

Example 2.2.Express the functionin the form.

Solution.Using the elementary properties of complex numbers, it follows that

so that.

Examples 2.1 and 2.2 show how to find u(x,y) and v(x,y) when a rule for computing f is given. Conversely, if u(x,y) and v(x,y) are two real-valued functions of the real
variables x and y, they determine a complex-valued function,and we can use the formulas

and

to find a formula for f involving the variables z and .

Example 2.3.Expressby a formula involving the variables.

Solution.Calculation reveals that

[Graphics:Images/ComplexFunLinear_gr_60.gif]

Usingin the expression of a complex function f may be convenient.It gives us the polar representation

,

where U and V are real functions of the real variables r and .

Remark.For a given function f, the functions u and v defined above are different from those used previously inwhich used Cartesian coordinates instead of polar coordinates.

Example 2.4.Expressin both Cartesian and polar form.

Solution.For the Cartesian form, a simple calculation gives

[Graphics:Images/ComplexFunLinear_gr_73.gif]

so that.

For the polar form, we get v

so that.

Remark.Once we have defined u and v for a function f in Cartesian form, we must use different symbols if we want to express f in polar form. As is clear here, the functions u and U are quite different, as are v and V.Of course, if we are working only in one context, we can use any symbols we choose.

For a given function f, the functions u and v defined here are different from those defined by equation (2-1), because equation (2-1) involves Cartesian coordinates and equation (2-2) involves polar coordinates.

Example 2.5.Expressin polar form.

Solution. We obtain

[Graphics:Images/ComplexFunLinear_gr_80.gif]

so that.

We now look at the geometric interpretation of a complex function.If D is the domain of real-valued functions u(x,y) and v(x,y), the equations

and

describe a transformation (or mapping) from D in the xy plane into the uv plane, also called the w plane. Therefore, we can also consider the function

to be a transformation (or mapping) from the set D in the z plane onto the range R in the w plane.This idea was illustrated in Figure 2.1. In the following paragraphs we present some additional key ideas. They are staples for any kind of function, and you should memorize all the terms in bold.

If A is a subset of the domain D of f, the setis called the image of the set A, and f is said to map A onto B.The image of a single point is a single point, and the image of the entire domain, D, is the range, R.The mappingis said to be from A into S if the image of A is contained in S.Mathematicians use the notation to indicate that a function maps A into S.Figure 2.2 illustrates a function f whose domain is D and whose range is R.The shaded areas depict that the function maps A onto B.The function also maps A into R, and, of course, it maps D onto R.

[Graphics:Images/ComplexFunLinear_gr_100.gif]

Figure 2.2 maps A onto B; maps A into R.

The inverse image of a point w is the set of all points z in D such that.The inverse image of a point may be one point, several points, or nothing at all.If the last case occurs then the point w is not in the range of f.For example, if,the inverse image of the point is the single point , because,and is the only point that maps to .In the case of,the inverse image of the point is the set .You will learn in Chapter 5 that, if,the inverse image of the point 0 is the empty set---there is no complex number z such that .

The inverse image of a set of points, S, is the collection of all points in the domain that map into S.If f maps D onto R it is possible for the inverse image of R to be function as well, but the original function must have a special property: a function f is said to be one-to-one if it maps distinct pointsonto distinct points.Many times an easy way to prove that a function f is one-to-one is to suppose,and from this assumption deduce that must equal .Thus,is one-to-one because if ,then .Dividing both sides of the last equation by gives.Figure 2.3 illustrates the idea of a one-to-one function: distinct points get mapped to distinct points.

[Graphics:Images/ComplexFunLinear_gr_125.gif]

Figure 2.3A functionw = f(z)that is one-to-one.

The functionis not one-to-one because ,but.Figure 2.4 depicts this situation: at least two different points get mapped to the same point.

[Graphics:Images/ComplexFunLinear_gr_129.gif]

Figure 2.4A function that is not one-to-one.

In the exercises we ask you to demonstrate that one-to-one functions give rise to inverses that are functions.Loosely speaking, ifmaps the set A one-to-one and onto the set B, then for each w in B there exists exactly one point z in AA such that.For any such value of z we can take the equationand "solve" for z as a function of w.Doing so produces an inverse functionwhere the following equations hold:

[Graphics:Images/ComplexFunLinear_gr_134.gif]

Conversely, if and are functions that map A into B and B into A, respectively, and the above hold, then f maps the set A one-to-one and onto the set B.

Further, if f is a one-to-one mapping from D onto T and if A is a subset of D, then f is a one-to-one mapping from A onto its image B.We can also show that, ifis a one-to-one mapping from A onto B andis a one-to-one mapping from B onto S, then the composite mappingis a one-to-one mapping from A onto S.

We usually indicate the inverse of by the symbol .If the domains of and are A and B respectively, then we write

for all,and

for all.

Also, forand.

iff,and

iff.

Next>>

Discussion Center

Discuss/
Query
Papers/
Syllabus
Feedback/
Suggestion
Yahoo
Groups
Sirfdosti
Groups
Contact
Us

MEMBERS LOGIN

INTERVIEW EBOOK

Get 9,000+ Interview Questions & Answers in an eBook. Interview Question & Answer Guide

9,000+ Interview Questions
All Questions Answered
5 FREE Bonuses
Free Upgrades

GATE RESOURCES

Gate Books

Training Institutes

Gate FAQs

GATE BOOKS

Mechanical Engineeering Books

Robotics Automations Engineering Books

Civil Engineering Books

Chemical Engineering Books

Environmental Engineering Books

Electrical Engineering Books

Electronics Engineering Books

Information Technology Books

Software Engineering Books

GATE Preparation Books

Exciting Offers

GATE Exam, Gate 2009, Gate Papers, Gate Preparation & Related Pages GATE Overview \| GATE Eligibility \| Structure Of GATE \| GATE Training Institutes \| Colleges Providing M.Tech/M.E. \| GATE Score \| GATE Results \| PG with Scholarships \| Article On GATE \| GATE Forum \| GATE 2009 Exclusive \| GATE 2009 Syllabus \| GATE Organizing Institute \| Important Dates for GATE Exam \| How to Apply for GATE \| Discipline / Branch Codes \| GATE Syllabus for Aerospace Engineering \| GATE Syllabus for Agricultural Engineering \| GATE Syllabus for Architecture and Planning \| GATE Syllabus for Chemical Engineering \| GATE Syllabus for Chemistry \| GATE Syllabus for Civil Engineering \| GATE Syllabus for Computer Science / IT \| GATE Syllabus for Electronics and Communication Engineering \| GATE Syllabus for Engineering Sciences \| GATE Syllabus for Geology and Geophysics \| GATE Syllabus for Instrumentation Engineering \| GATE Syllabus for Life Sciences \| GATE Syllabus for Mathematics \| GATE Syllabus for Mechanical Engineering \| GATE Syllabus for Metallurgical Engineering \| GATE Syllabus for Mining Engineering \| GATE Syllabus for Physics \| GATE Syllabus for Production and Industrial Engineering \| GATE Syllabus for Pharmaceutical Sciences \| GATE Syllabus for Textile Engineering and Fibre Science \| GATE Preparation \| GATE Pattern \| GATE Tips & Tricks \| GATE Compare Evaluation \| GATE Sample Papers \| GATE Downloads \| Experts View on GATE \| CEED 2009 \| CEED 2009 Exam \| Eligibility for CEED Exam \| Application forms of CEED Exam \| Important Dates of CEED Exam \| Contact Address for CEED Exam \| CEED Examination Centres \| CEED Sample Papers \| Discuss GATE \| GATE Forum of OneStopGATE.com \| GATE Exam Cities \| Contact Details for GATE \| Bank Details for GATE \| GATE Miscellaneous Info \| GATE FAQs \| Advertisement on GATE \| Contact Us on OneStopGATE \|
Copyright © 2024. One Stop Gate.com. All rights reserved	Testimonials \|Link To Us \|Sitemap \|Privacy Policy \| Terms and Conditions\|About Us
Our Portals : Academic Tutorials \| Best eBooksworld \| Beyond Stats \| City Details \| Interview Questions \| India Job Forum \| Excellent Mobiles \| Free Bangalore \| Give Me The Code \| Gog Logo \| Free Classifieds \| Jobs Assist \| Interview Questions \| One Stop FAQs \| One Stop GATE \| One Stop GRE \| One Stop IAS \| One Stop MBA \| One Stop SAP \| One Stop Testing \| Web Hosting \| Quick Site Kit \| Sirf Dosti \| Source Codes World \| Tasty Food \| Tech Archive \| Software Testing Interview Questions \| Free Online Exams \| The Galz \| Top Masala \| Vyom \| Vyom eBooks \| Vyom International \| Vyom Links \| Vyoms \| Vyom World C Interview Questions \| C++ Interview Questions \| Send Free SMS \| Placement Papers \| SMS Jokes \| Cool Forwards \| Romantic Shayari