Complex Integrals

According to Churchill’s book Complex Variables and Applications,

Integrals are extremely important in the study of functions of a complex variable. The theory of integration, to be developed in this chapter, is noted for its mathematical elegance. The theorems are generally concise and powerful, and many of the proofs are short.

Basic Contour Integrals

The basic way of computing contour integrals is to use the definition. There are more advanced and very powerful methods of computing contour integrals, which we will mention in later posts.

The summarised definition is as follows: \int_C f(z)\ dz=\int_a^b f[z(t)]z'(t)\,dt where z=z(t), a\leq t\leq b represents a contour C.

Basic Example 1: I=\int_C \bar{z}\,dz, where C is the contour z=2e^{i\theta}, -\pi/2\leq\theta\leq\pi/2.

Using the definition, we have

\begin{aligned}    I&=\int_{-\pi/2}^{\pi/2}2e^{-it}\cdot 2ie^{it}\,dt\\    &=4i\int_{-\pi/2}^{\pi/2} 1\,dt\\    &=4\pi i    \end{aligned}

Advertisements

About mathtuition88

http://mathtuition88.com
This entry was posted in math and tagged . Bookmark the permalink.

2 Responses to Complex Integrals

  1. nkpithwa says:

    Elementary complex math is also beautifully explained in Advanced Engineering Mathematics — by Erwin Kreyszig. Also, complex variable techniques are fully exploited by engineers esp. electrical, mechanical, chemical…I do not know about other engineering disciplines. But, yes, as far as first course in complex variables Is concerned, for pure/applied mathematicians, Churchill’s book is the best

    Liked by 1 person

    • Thanks for recommending the book by Kreszig. I read his excellent book on Functional Analysis. Churchill’s book has a lot of diagrams, which is good for beginners trying to learn the theory.

      Like

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s