hw0 - Math 260, Spring 2012 Jerry L. Kazdan Homework Set 0...

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Unformatted text preview: Math 260, Spring 2012 Jerry L. Kazdan Homework Set 0 [Due: Never] Comples Power Series In our treatment of both differential equations and Fourier series, it will be essential to use comples numbers and complex power series. They enormously simplify the story. This is treated in the Lecture Notes http://hans.math.upenn.edu/ kazdan/260S12/notes/math21/math21-2011.pdf , Chapter 0.7, Chapter 1.6. The guiding force is Eulers use of power series to define e z , where z = x + iy is a complex number by using an analogy with the real power series. Thus, he wrote e z := 1 + z + z 2 2! + + z k k ! + = X k =1 z k k ! . He might have been led to this by observing the close relatonship between the familiar real power series e x = 1 + x + x 2 2! + x 3 3! + + x k k ! + + cos x = 1- x 2 2! + x 4 4! + (- 1) k x 2 k (2 k )! + sin x = x- x 3 3! + x 5 5! + + (- 1) k +1 x 2 k +1 (2 k + 1)! + Thus, e x has all the powers, cos x only the even powers, and sin x only the odd powers. The denominators match-up but then the alternating signs seem mysterious. The mystery symplifies dramatically because Euler had the courage to utilize the sign pattern i = 1 , i 1 = i, i 2 =- 1 , i 3 =- i, i 4 = 1 , i 5 = i, . . . . Then comparing power series we see the remarkable fit in Eulers extrordinary formula e it = cos t + i sin t. (1) From this all the properties of the trigonometric functions are consequences of the simpler e z + w = e z e w . (2) Example We derive the usual formulas for cos(...
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hw0 - Math 260, Spring 2012 Jerry L. Kazdan Homework Set 0...

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