Call Now to Set Up Tutoring:

(888) 888-0446

Complex Analysis : Complex Integration

Study concepts, example questions & explanations for Complex Analysis

Create An Account

All Complex Analysis Resources

1 Diagnostic Test 13 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Complex Integration

Let be the region of the complex plane enclosed by $x = \pm 2 \text{ and } y = \pm 2$

Compute $\int_C \frac{\cos z}{z(z^2+8)} dz$

Possible Answers:

$\frac{\pi i }{2}$

$\frac{\pi i }{4}$

$\frac{\pi i}{8}$

$2 \pi i$

$\pi i$

Correct answer:

$\frac{\pi i }{4}$

Explanation:

Recall Cauchy's integral formula, which states

$f(z) = \frac{1}{2\pi i}\int_C\frac{f(s)}{s-z}ds$

In this case, we have $f(s) = \frac{cos(s)}{s^2 + 8}$

Plugging in gives us

$\int_C \frac{\frac{\cos z}{z^2+8}}{z} dz = 2\pi i f(0) = 2\pi i \frac{1}{8} = \frac{\pi i }{4}$

Report an Error

Example Question #2 : Complex Integration

Let be the unit circle.

Compute $\int_C z^2 sin\left(\frac{1}{z}\right) dz$

Possible Answers:

$-\frac{\pi i}{6}$

$-\frac{\pi i}{3}$

$-\frac{\pi i}{2}$

$-\pi i$

Correct answer:

$-\frac{\pi i}{3}$

Explanation:

Recall the Taylor expansion of

$\sin(z) = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + ...$

Use this to write

$z^2\sin\left(\frac{1}{z}\right) = z - \frac{1}{z3!} + \frac{1}{z^35!} - \frac{1}{z^57!} + ...$

The coefficient of $\frac{1}{z}$ is the residue, so by the residue theorem, the value of the integral is

$-\frac{\pi i}{3}$

Report an Error

View Tutors

Teagan
Certified Tutor

The University of Texas at Austin, Bachelor of Science, Physics.

View Tutors

Bryan
Certified Tutor

View Tutors

Blanca
Certified Tutor

University of Phoenix-Tucson Campus, Bachelor of Science, Business Administration and Management.

All Complex Analysis Resources

1 Diagnostic Test 13 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Los Angeles, Calculus Tutors in Atlanta, Physics Tutors in Phoenix, GRE Tutors in Seattle, SSAT Tutors in Seattle, Calculus Tutors in Phoenix, Chemistry Tutors in Boston, Physics Tutors in Miami, Chemistry Tutors in San Francisco-Bay Area, GMAT Tutors in Houston

Popular Courses & Classes

Spanish Courses & Classes in Houston, MCAT Courses & Classes in Atlanta, SSAT Courses & Classes in San Diego, SSAT Courses & Classes in Denver, ACT Courses & Classes in San Diego, ISEE Courses & Classes in Denver, GMAT Courses & Classes in Seattle, GRE Courses & Classes in Houston, SSAT Courses & Classes in Boston, SAT Courses & Classes in San Francisco-Bay Area

Popular Test Prep

GMAT Test Prep in Denver, GRE Test Prep in Houston, MCAT Test Prep in Denver, ISEE Test Prep in Houston, SSAT Test Prep in Chicago, SAT Test Prep in Miami, GMAT Test Prep in Seattle, SAT Test Prep in Denver, GRE Test Prep in Washington DC, GRE Test Prep in Seattle

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Complex Analysis : Complex Integration

Study concepts, example questions & explanations for Complex Analysis

All Complex Analysis Resources

Example Questions

Example Question #1 : Complex Integration

Example Question #2 : Complex Integration

All Complex Analysis Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: