Formula2

Formula2 - Math 421:01 and 02 — Spring 2008 Second Review...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 421:01 and 02 — Spring 2008 Second Review and Formula Sheet Trigonometry The pair . cos t; sin t/ gives the parametrization of the unit circle by arc length starting from the positive x axis. Thus, these functions are periodic with period 2 and some special values are: cos D 1 , sin D , cos =2 D , sin =2 D 1 . Also, cos .x C / D cos x and sin .x C / D sin x . The formula e it D cos t C i sin t (with i 2 D 1 ) connects exponential and trigonometric functions. It can be used to relate the trigonometric form of a Fourier series to the complex form of the series. Consequences include the basic identities cos .˛ C ˇ/ D cos ˛ cos ˇ sin ˛ sin ˇ sin .˛ C ˇ/ D sin ˛ cos ˇ C cos ˛ sin ˇ that are used to prove the orthogonality of the functions f 1;. cos nx; sin nx W n D 1;2;:::/ g on the interval < x < . Ordinary Differential Equations There are two families of differential equations that are easy to solve exactly . ) Linear equations with constant coefficients . Usually, the solutions have the form y D e cx . . ) Euler equations , whose terms are combinations with constant coefficients of x k d k y=dx k for several integer k . Usually, the solutions have the form y D x c . To solve these equations, try the correct form of solution. The equation reduces to an algebraic equation in c . If a root c is complex, the exponential is usually converted to a form containing trigonometric functions. For Euler equations, this also requires the identity x c D e c ln x . If the algebraic equation has repeated roots , this method does not give enough solutions. If c is a repeated root, a second solution is xe cx in the constant coefficient case, or x c ln x in the Euler equation case. A full characterization of solutions will not be given here, since it will be easy to check any tentative solution constructed from this principle. Fourier series The Fourier series of a function f.x/ on the interval p < x < p is given by a 2 C 1 X n D 1 a n cos n x p C b n sin n x p where a D 1 p Z p p f.x/dx a n D 1 p Z p p f.x/ cos n x p dx b n D 1 p Z p p f.x/ sin n x p dx . / The constant term a =2 is the average value of f.x/ , and Z p p cos 2 n x p dx D p Z p p sin 2 n x p dx D p so that the Fourier series of f.x/ D 1 , f.x/ D cos n x=p or f.x/ D sin n x=p is a single term identical to f.x/ . Math 421:01 and 02, April 09, 2008, p. 2 Note that the functions cos n x=p and sin n x=p are periodic with period 2p . When computing a Fourier series, the description of a function should be thought of as representing a single period of a function with period length 2p ....
View Full Document

This note was uploaded on 09/29/2009 for the course 650 421 taught by Professor Bumby during the Spring '08 term at Rutgers.

Page1 / 5

Formula2 - Math 421:01 and 02 — Spring 2008 Second Review...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online