×

Let's log you in.

or

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

×

or

by: Braeden Lind

8

0

6

Introduction to Applied Mathematics MA 325

Marketplace > North Carolina State University > Mathematics (M) > MA 325 > Introduction to Applied Mathematics
Braeden Lind
NCS
GPA 3.93

Robert White

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.
Robert White
TYPE
Class Notes
PAGES
6
WORDS
KARMA
25 ?

Popular in Mathematics (M)

This 6 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 325 at North Carolina State University taught by Robert White in Fall. Since its upload, it has received 8 views. For similar materials see /class/223679/ma-325-north-carolina-state-university in Mathematics (M) at North Carolina State University.

×

Reviews for Introduction to Applied Mathematics

×

×

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: 10/15/15
Lecture 5 Complex Numbers and Fourier Transforms The imaginary number i sqrt1 is not a real number because 1392 1 is not positive The complex numbers are the set whose elements a bi where a and b are any real numbers called the real and imaginary parts of a bi One can view them as vectors in the complex plane where a is on the horizontal axis and b is on the vertical axis Then define addition of two complex numbers as vector addition the modulus of the complex number as the length of the vector and the conjugate of the complex number as a complex number corresponding to the re ection about the horizontal axis a ci bd cid modulus ofaib sq rtaquot2bquot2 The product and division of numbers are de ned by using 1392 1 aibcid E ac bd iadbc and aibcid E aibcid cidcid ac bd c2 612 iadbc c2 612 Using these de nitions one can compute more complicated functions of complex variables A very important function is the exponential function and we use its series representation e lz39x1l ix22ix33ix44 lix22ix44 z39x1l my3 cosx 139 sinx Euler39sFormula The modulus of e is equal to one because sin2x 0032x 1 Also x can be viewed as the angle from the real aXis to the vector representing the complex number e i2 7rn i2 7quot An important case is when x 27W so that w Ee and z E w e satisfy w lz wk w39kcmallww2wquot391 0 The last property is established by using wquot 1 and noting w is not equal to l wlww2 w 391 ww2 wquot ww2 wquot391 1 w ll w w2 w 030 thatforw lnot equal tozero lww2wquot3910 The compleX numbers wj with j 0n 1 are called the rim roots of unity because they solve x 1 For n 3 they the following vectors whose vector sum is zero ei21t32 Application of Complex Numbers to Laplace Transform Consider the Laplace transform if the complex exponential function em Here we make the natural extension to complex integrals By Euler s formula Lg Lc0sbt i Sinbt Lc0sbtiLSinbt39 Apply the definition of Laplace transform directly to em 81172 J39ers texbtdt 0 N limNgoo Ie39ste b39dt 0 N limmw j army 0 lim ersxl7N 1 ersxl70 1 NT sz39b sz39b 1 s ib 1 sib s ib sib s 1 32 b2 32 b2 By equating the real and imagery parts of Leibt we have derived the Laplace transforms for the sine and cosine function These rules could also have been derived by the direct application of the definition of the Laplace transform to the sine and cosine functions The Fourier Transform The Fourier transform of a function ft could be viewed as a variation on the Laplace transform where the real parameter s is replaced by an imaginary parameter iw and the domain of integration is extended to the entire real line Def39mtion The Fourier transform off is Ff j equot ftdt Basic Properties of Fourier Transform FCf CF Ff g Ff Fg Ff z39wFfand Ffg FfFgwhere fg j frgtrdr 59 H Application to Solution of uxx cu 139 Replace the variable t with x and compute the Fourier transform of both sides in the differential equation By using the third property the derivative property twice we obtain FumcuFf ia2Fu cFu F f W2 cFuFf Fu Ff 1 w2c Apply the fourth property the convolution property to conclude ufg where Fg a22c Discrete Fourier Transform The discrete Fourier transform is derived by truncating the integral to 0 1 replacing w by 27k and letting fjn The discrete Fourier transform of f fg H124 is complex vector whose k1h component is r171 r171 ea27rkjnf Iannle z j0 Another way to view this is as a matrix product from vectors in real n dimensional space to vectors in complex n dimensional space Ff where F has kj component 21 For example ifn 3 then 20 20 20 l l 1 F 0 1 2 1 21 22 0 22 4 1 22 1 Iff 1 72 n 3 and2 e42 12 132139 then 1 1 1 1 10 10 Ff 1 21 22 7 l72222 350 433i 2 l72222 350433i 122 21 In applications 71 is typically very large and an efficient method to do these computations is required In Matlab the command fft is an implementation of the fast Fourier transform This requires W2log2n operations which is much less than the 2112 operations for the matrixvector product The following are some simple examples gtgtfft172 n3 100 35000 43301i 35000 43301i Matlab Code fftt gm t 00011 n 1001 freq 111001 fftsin f 4sin2pi40t fftcos f 8cos2pi100t subp10t211 p10tfreqrealf sinfreqrea1f cos subp10t2 12 p10tfreqimagf sinfreqimagf cos The spikes in the Fourier transform plots identify the frequencies f and n f 4000 t real t cos 3000 0 2000 7 1000 1000 W 600 800 W 1000 frequency 1200 2000 t t 10007 J imaQUTKSinOD JI 0 V 1000 7 2000 W 600 800 W 1000 frequency 1200

×

25 Karma

×

×

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

Jim McGreen Ohio University

"Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

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