### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Class Note for MATH 322 with Professor Glickenstein at UA

### View Full Document

## 14

## 0

## Popular in Course

## Popular in Department

This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 14 views.

## Popular in Subject

## Reviews for Class Note for MATH 322 with Professor Glickenstein at UA

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 02/06/15

Chapter 13 Complex Numbers Sections 135 136 amp 137 Chapter 13 CmnplE Numbers Dehmuon tiea o The exponential of a complex number 2 x iy is defined as expz eXPXy eXPXeXPiy expX COW 139 siny 0 As for real numbers the exp derivative ie dz 0 The exponential is therefore onential function is equal to its 1 expz expz 1 entire 9 You may also use the notation expz ez Chapter 13 CmnplE Numbers o The exponential function is periodic with period 27ri indeed for any integer k E Z expz 2k7ri expx cosy 2k7r isiny 2k7r expx cosy isiny expz 0 Moreover leXPZl leXPXl leXP l eXPX COSZM sin2y expx exp lRez a As with real numbers a exp21 22 exp21 exp22 o expz 75 0 Chapter 13 CmnplE Numbers Trigonomelrlc and b C 2 Trigonometric functions 0 The complex sine and cosine functions are defined in a way similar to their real counterparts eiz eiiz cosz 2 IZ eilz sinz 2 o The tangent cotangent secant and cosecant are defined as usual For instance sinz tanlz cosz7 secz L etc cosz Chapter 13 CmnplE Numbers 1 quotemal Trigonomelrlc and h Tn gonom emc functions vaerholln Inntiuns Trigonometric functions continued 0 The rules of differentiation that you are familiar with still work 0 Example 9 Use the definitions of cosz and sinz eiz eiiz Biz 2 7 to find cosz and sinz cosz a Show that Euler s formula also works if 0 is complex c fl I nc non IS functions I lunctu en Hyerbolic functions o The complex hyperbolic sine and similar to their real counterparts cosine are defined in a way eZ e Z Z Z 2 7 sinhz e e coshz 3 o The hyperbolic sine and cosine as well as the sine and cosine are entire 0 We have the following relations coshiz cosz7 sinhiz i sinz7 4 cosiz coshz7 siniz i sinhz on 939 Nu m b ers Chapter CmnpleyNunibers Chapter a Dennmon pnm Ipal value 2 M iii 4 Complex logarithm o The logarithm w of 2 0 is defined as eW z 0 Since the exponential is 27ri periodic the complex logarithm is multi valued 0 Solving the above equation for w w M and z rem gives W 7 eW eW elW39 rem gt e r T r 7 w 0 2p7r which implies w lnr and w 0 2p7r p E Z 0 Therefore lnz lnlzl i argz 7 ple Numbers Chapter reinn Ion prlnclpal value of lnl lzll Principal value of lnz 0 We define the principal value of lnz Lnz as the value of lnz obtained with the principal value of argz ie Lnz lnlzl I39Argz 0 Note that Lnz jumps by 727d when All312 one crosses the 0 X negative real axis Argz gtTc from above 0 The negative real axis is called a branch cut of Lnz 939 Numbers Definition Principal value of im lzll 3 Recall that Lnz lnlzl I39Argz 0 Since Argz argz 2p7r p E Z we therefore see that lnz is related to Lnz by lnz Lnz i2p7r p E Z 0 Examples 0 Ln2 ln2 but ln2 ln2 i2p7r7 c Find Ln74 and ln74 0 Find ln10i pEZ Chapter 13 Compier Numbers Deiuu Ion prlnclpal value of lnl lzll Tm lunctu en Properties of the logarithm a You have to be careful when you use identities like 21 ln2122 ln21ln22 or In lnzliln22 They are only true up to multiples of 27ri o For instance if 21 i expi7r2 and 22 71 expi7r ln21 i 2p1i7r ln22 i7r2p2i7r p1p2 E Z and 3 ln2122 i rls2p3i7r7 pg 62 but p3 is not necessarily equal to p1 p2 Chapter 13 umpier Numbers Definition Principal value of im lzll continued 0 Moreover with 21 i expi7r2 and 22 71 expi7r Ln21 ig Ln22 Mr and Ln21 22 7 y Ln21 Ln22 0 However every branch of the logarithm ie each expression of lnz with a given value of p E Z is analytic except at the branch point z 0 and on the branch cut of lnz In the domain of analyticity of lnz d 1 Ellnlzll 2 Chapter 13 Compier Numbers 5 Definition ump r um n 5 Complex power function o lf 2 0 and c are complex numbers we define zc expclnz exp anz2pc7r p62 0 For c E C this is again a multi valued function and we define the principal value of 25 as zc exp c Lnz Chapter 13 umpier Numbers

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over $600 per month. I LOVE StudySoup!"

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.