A Quick Introduction to Complex Analysis
eBook - ePub

A Quick Introduction to Complex Analysis

Kalyan Chakraborty, Shigeru Kanemitsu;Takako Kuzumaki

Share book
  1. 208 pages
  2. English
  3. ePUB (mobile friendly)
  4. Available on iOS & Android
eBook - ePub

A Quick Introduction to Complex Analysis

Kalyan Chakraborty, Shigeru Kanemitsu;Takako Kuzumaki

Book details
Book preview
Table of contents

About This Book

The aim of the book is to give a smooth analytic continuation from calculus to complex analysis by way of plenty of practical examples and worked-out exercises. The scope ranges from applications in calculus to complex analysis in two different levels.

Frequently asked questions

How do I cancel my subscription?
Simply head over to the account section in settings and click on “Cancel Subscription” - it’s as simple as that. After you cancel, your membership will stay active for the remainder of the time you’ve paid for. Learn more here.
Can/how do I download books?
At the moment all of our mobile-responsive ePub books are available to download via the app. Most of our PDFs are also available to download and we're working on making the final remaining ones downloadable now. Learn more here.
What is the difference between the pricing plans?
Both plans give you full access to the library and all of Perlego’s features. The only differences are the price and subscription period: With the annual plan you’ll save around 30% compared to 12 months on the monthly plan.
What is Perlego?
We are an online textbook subscription service, where you can get access to an entire online library for less than the price of a single book per month. With over 1 million books across 1000+ topics, we’ve got you covered! Learn more here.
Do you support text-to-speech?
Look out for the read-aloud symbol on your next book to see if you can listen to it. The read-aloud tool reads text aloud for you, highlighting the text as it is being read. You can pause it, speed it up and slow it down. Learn more here.
Is A Quick Introduction to Complex Analysis an online PDF/ePUB?
Yes, you can access A Quick Introduction to Complex Analysis by Kalyan Chakraborty, Shigeru Kanemitsu;Takako Kuzumaki in PDF and/or ePUB format, as well as other popular books in Mathematics & Complex Analysis. We have over one million books available in our catalogue for you to explore.



Chapter 1

A Quick Introduction to Complex Analysis with Applications

1.1The quickest introduction to complex analysis

This section serves as a review or survey material for those who are in a hurry to recall what the main ingredients in complex analysis are in three days. It presupposes some basic knowledge of complex numbers and functions etc. If you encounter something whose meaning is not clear to you, you must make a recourse to a more standard presentation of the theory in the subsequent sections.
Since complex function theory is in a sense infinitesimal calculus of complex-valued functions in the complex variable, it follows that the most fundamental ingredients in the theory are differentiation and integration; the former of which we define in the same way as with real functions, i.e. the function f(z) is differentiable at z = z0 in its domain of definition D if
exists, in which case, the limit value is denoted by f′(z0) called the derivative of f at z0. Formally this is the same as the derivative of a real function, the only difference being that the limit is taken in a planar domain D ⊂ ℂ.
On the other hand, the latter is more complicated since integration means a contour integral. However, the below-mentioned “performing the change of variable principle” makes it quite easily accessible.
Definition 1.1. We say that a complex function is analytic (or sometimes holomorphic or regular) in a domain if it is differentiable at each point of the domain, where differentiability means the existence of the derivative, and a domain (sometimes referred to as a region) mathematically means a connected set; we simply understand a domain to be a certain plane figure ( ℂ) with interior and with the boundary curve. We usually assume that domains are arc-wise connected. Typical domains are the rectangles (parallelopipeds) and circles and there is no need to worry about what domains are.
We assume throughout that a curve is a piecewise smooth (Jordan) curve described by the parametric expression
E.g., the unit circle C: ∣z∣ = 1 is given by
or by
C: z = z(t) = eit, t ∈ [0, 2π],
where the complex exponential function eit, cf. §1.3.2.
By the Jordan curve theorem, such a curve encircles a ...

Table of contents