×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

13

0

4

# Class Note for MATH 583A with Professor Faris at UA

Marketplace > University of Arizona > Class Note for MATH 583A with Professor Faris at UA

No professor available

These notes were just uploaded, and will be ready to view shortly.

Either way, we'll remind you when they're ready :)

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

COURSE
PROF.
No professor available
TYPE
Class Notes
PAGES
4
WORDS
KARMA
25 ?

## Popular in Department

This 4 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 13 views.

×

## Reviews for Class Note for MATH 583A with Professor Faris at UA

×

×

### What is Karma?

#### 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
Radial functions and the Fourier transform Notes for Math 583A7 Fall 2008 December 67 2008 1 Area of a sphere The volume in n dimensions is vol dnx dzl dzn T7 1 dT dnilwi 1 Here T lxl is the radius7 and w xT it a radial unit vector Also dn lw denotes the angular integral For instance7 when n 2 it is di for 0 S 9 S 2 while for n 3 it is sini9 dgdqb for 0 S 9 S 7r and 0 S 45 S 27L The radial component of the volume gives the area of the sphere The radial directional derivative along the unit vector w xT may be denoted 1 8 8 8 wd7zlaixlznmigi 2 The corresponding spherical area is wdlvol T7 1 dnilwi 3 Thus when n 2 it is lT I dyiydz T d9 while for n 3 it is lT I dy dz y d2 dz 1 dz dy 7 2 sini9 d9 dqbi The divergence theorem for the ball Br of radius T is thus div vdnx v w Tn ldn lwi 4 BT T Notice that if one takes V X7 then div x n7 while x w T This shows that n times the volume of the ball is T times the surface area of the sphere Recall that the Gamma function is de ned by Fz fem tzequoti It is easy to show that Fz 1 Since lquotl 17 it follows that Fn n 7 UL The result 7r follows reduction to a Gaussian integrali It follows that re a Theorem 1 The aTea 0f the unit spheTe Sn1 Q R is Wnil lt5 Thus in 3 dimensions the area of the sphere is mg 4 While in 2 dimensions the circumference of the circle is wl 27L ln 1 dimension the two points get count we 2 To prove this theorem7 consider the Gaussian integral 2 e dnx 1A 6 Rn ln polar coordinates this is n 00 T2 wn127r77 e Trn 1 dr ll 7 0 Let u r22i Then this is n ni2 Do n72 wn127r772T e uuT du ll 8 0 That is n wn17r7271r 1i 9 This gives the result 2 Fourier transform of a power Theorem 2 Let 1 lt a lt n The Fourier transform of llzl is Calkln where 0a 2wi 10 This is not too dif cult It is clear from scaling that the Fourier transform of llzl is Clkln T It remains to evaluate the constant C Take the inner product With the Gaussian This gives n x2 l x2 l 27r 7e7T a dnx 2W ne TC nia dnki 11 n W Rquot W Writing this in polar coordinates gives 2W e TTzrniliu dr C27r n e TTgrliu dri 12 0 0 This in turn gives 2 2quot 2r a C2 n2 THE 13 3 The Hankel transform De ne the Bessel function Jyt t WSW sumo 491 14 7w 270 1 2V 0 This makes sense for all real numbers 1 2 0 but we shall be interested mainly in the cases When 1 is an integer or 1 is a halfinteger In the case When 1 is a halfinteger the exponent 21 is odd and so it is possible to evaluate the integral in terms of elementary functions Thus for example it 1 2 ti sint 2 em 2 270 t This is not possible When 1 is an integer Thus for 1 0 we have the relatively mysterious expression J1t 7r e i smw sin19 d9 15 0 J0t i 0 em 49 16 Fix a value of 11 If we consider a function gr its Hankel transform is the function y 8 given by s 000 Jysrgrrdr1 17 We shall see that the Hankel transform is related to the Fourier transforml 4 The radial Fourier transform The rst result is that the radial Fourier transform is given by a Hankel trans formi Suppose f is a function on Rni lts Fourier transform is fk e ik39x x anxi 18 Let T x and s k Write fx FT and fk Fns Theorem 3 The radial Fourier transform in n dimensions is given in terms of the Hankel transform by 3T2Fns 270 Am JT2STTquotT 2FTMT1 19 Here is the proof of the theoremi lntroduce polar coordinates With the 2 axis along k so that k x sr cos191 Suppose that the function is radial that is Fr1 fk 39ns OooAWe iSTC s9Frwn2sin19 2dt9r 1dr1 20 3 Use n72 tT WSW sin 9 2 49 21 For the case n 3 the Bessel function has order 12 and has the above expression in terms of elementary functions So F3s 477 Am Mpg dr 22 87 For n 2 the Bessel function has order 0 We get 523 277 Am J05TFTT dr 23

×

×

### BOOM! Enjoy Your Free Notes!

×

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

Steve Martinelli UC Los Angeles

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

Allison Fischer University of Alabama

#### "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!"

Bentley McCaw University of Florida

Forbes

#### "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."

Become an Elite Notetaker and start selling your notes online!
×

### 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