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

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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 0 D 1 , sin 0 D 0 , cos =2 D 0 , 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 0 2 C 1 X n D 1 a n cos n x p C b n sin n x p where a 0 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 0 =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/ .
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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 . If f .x/ is defined by cases , integrals containing f .x/ should be split into integrals over the several intervals where f .x/ may be replaced by an expression used in the definition.
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