# hwk7 - e z = e x(cos y i sin y as stated in class(iv We...

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Math 31BH - Homework 7. Due Thursday, March 3. 1. From the textbook, solve 2 . 10 . 2 , 2 . 10 . 5 , 2 . 10 . 9 , 2 . 10 . 12 . 2. Problems 3 . 1 . 2 , 3 . 1 . 5( a ) , 3 . 1 . 6 , 3 . 1 . 7 , 3 . 1 . 10 , 3 . 1 . 11 , 3 . 1 . 19( a ) . 3. Recall the Taylor expansion e z = 1 + z + z 2 2! + z 3 3! + .... (i) Formally substitute z = iy and ﬁnd the Taylor expansion of e iy . (ii) Find the Taylor expansions of cos y and sin y . (iii) Prove that e iy = cos y + i sin y and derive that
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Unformatted text preview: e z = e x (cos y + i sin y ) as stated in class. (iv) We showed in class that z → w = e z is locally invertible. Write down an inverse for the exponential valid in a neighborhood of w = πi 3 . Where is the local inverse (the logarithm) deﬁned? 1...
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