Worksheet 3 Solutions

You just have to keep track of the exponents eix cosx

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Unformatted text preview: ) and sin(x) into Euler’s formula and show that ∞ (ix)k . k! eix = k=0 The idea is to rewrite ±xk so that it looks like (ix)k . Use the fact that −1 = i2 , −i = i3 , 1 = i4 , i = i5 , and so on. Solution: Basically, this problem is all about bookkeeping. You just have to keep track of the exponents: eix = cos(x) + i sin(x). = 1− x2 x4 x6 + − + ··· 2! 4! 6! = 1+ (−1)x2 (1)x4 (−1)x6 + + + ··· 2! 4! 6! = 1+ (i2 )x2 (i4 )x4 (i6 )x6 + + + ··· 2! 4! 6! = 1+ (ix)2 (ix)4 (ix)6 + + + ··· 2! 4! 6! = 1 + ix + +i x− x3 x5 x7 + − + ··· 3! 5! 7! + ix + + ix + + ix + (−i)x3 ix5 (−i)x7 + + + ··· 3! 5! 7! (i3 )x3 (i5 )x5 (i7 )x7 + + + ··· 3! 5! 7! (ix)3 (ix)5 (ix)7 + + + ··· 3! 5! 7! (ix)2 (ix)3 (ix)4 (ix)5 (ix)6 (ix)7 + + + + +...
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