Call Now to Set Up Tutoring:

(888) 888-0446

Complex Analysis : Complex Functions

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 #21 : Complex Analysis

What does the sum below equal?

$\sum_{n=1}^{m}(e^{2\pi i/m})^{n}$

Another way of asking this question is what is the sum of the roots of unity.

$m\in \mathbb{N}$

Possible Answers:

$\pi$

$e^\pi$

Correct answer:

Explanation:

As messy as it looks, this is just a geometric series.

we will use the partial sum formula for the geometric series.

$S_{n}=\frac{a_{1}(1-r^n)}{1-r}$

$a_{1}=e^{2\pi i/m}$

$r=e^{2\pi i/m}$

$S_{m}=\frac{a_{1}(1-r^m)}{1-r}=\frac{e^{2\pi i/m}(1-{\color{Red} (e^{2\pi i/m})^m})}{1-e^{2\pi i/m}}$

the red part is the only part that matters....the m's cancel out leaving....

$S_{m}=\frac{a_{1}(1-r^m)}{1-r}=\frac{e^{2\pi i/m}(1-{\color{Red} (e^{2\pi i})})}{1-e^{2\pi i/m}}$

and...

$e^{2\pi i}=1$

thus we have...

$S_{m}=\frac{a_{1}(1-r^m)}{1-r}=\frac{e^{2\pi i/m}(1-{\color{Red} (1)})}{1-e^{2\pi i/m}}$

which gives the answer of zero.

$S_{m}=0$

Report an Error

Example Question #22 : Complex Analysis

Consider the function f(z) = |z|^2

Find an expression for f'(z) (hint: use the definition of derivatve) and where it exists in the complex plane.

Possible Answers:

$f'(z) = 2|z| \text{ where } z=0$

$f'(z) = 2|z| \text{ } \forall z$

$f'(z) = 2z \text{ } \forall z$

$f'(z) = 2z \text{ where } z=0$

$f'(z) = 0 \text{ where } z=0$

Correct answer:

$f'(z) = 0 \text{ where } z=0$

Explanation:

Applying the definition of derivative, we have that

$f'(z) = \lim_{\Delta z \to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z} \\$

$f'(z) = \lim_{\Delta z \to 0}\frac{|z+\Delta z|^2 - |z|^2}{\Delta z} = \lim_{\Delta z \to 0} \frac{(z+\Delta z)(\bar{z}+\overline{\Delta z}) - z\overline{z}}{\Delta z} = \lim_{\Delta z \to 0} \overline{z} + \overline{\Delta z} + z \frac{\overline{\Delta z}}{\Delta z}$

If the limits exists, it can be found by letting $\Delta z$ approach in any manner.

In particular, if we it approach through the points $(\Delta x , 0 )$ , we have that

$\overline{\Delta z} = \overline{\Delta x + i0} = \Delta x - i0 = \Delta x + i0 = \Delta z$

A similar approach with $(0, \Delta y)$ implies that $\overline{\Delta z } = -\Delta z$

Since limits are unique, these two approaches imply that

$\overline{z} + z = \overline{z} - z$ , which implies z=0 and f'(z) cannot exist when $z \neq 0$

Report an Error

View Tutors

Lillian
Certified Tutor

Smith College, Bachelor in Arts, History.

View Tutors

Robert
Certified Tutor

University of Colorado Boulder, Bachelor of Science, Biology, General. Colorado State University-Fort Collins, Doctor of Vete...

View Tutors

Moira
Certified Tutor

Georgetown University, Bachelor of Science, French. Temple University, Masters in Education, Special Education.

All Complex Analysis Resources

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

Popular Subjects

Physics Tutors in Denver, GRE Tutors in Houston, LSAT Tutors in Seattle, GRE Tutors in Philadelphia, English Tutors in Philadelphia, Reading Tutors in New York City, Computer Science Tutors in Miami, GMAT Tutors in Los Angeles, ACT Tutors in Dallas Fort Worth, Biology Tutors in New York City

Popular Courses & Classes

SSAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Washington DC, SAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Houston, GRE Courses & Classes in Houston, ACT Courses & Classes in New York City, MCAT Courses & Classes in Boston, ACT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in New York City

Popular Test Prep

ACT Test Prep in New York City, ACT Test Prep in Houston, ACT Test Prep in Dallas Fort Worth, ACT Test Prep in Los Angeles, GMAT Test Prep in Washington DC, SSAT Test Prep in Philadelphia, LSAT Test Prep in Miami, ISEE Test Prep in Washington DC, GMAT Test Prep in Boston, SAT Test Prep in Houston

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 Functions

Study concepts, example questions & explanations for Complex Analysis

All Complex Analysis Resources

Example Questions

Example Question #21 : Complex Analysis

Example Question #22 : Complex Analysis

All Complex Analysis Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: