Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

•• The function ez is defined for any z, real or complex, by its power series

Write down this series for the case that z is purely imaginary, z = iθ. Note that the terms in this series are alternately real and imaginary. Group together all the real terms and all the imaginary terms, and prove the important identity, called Euler’s formula,

 eiθ = cos θ + i sin θ (7.121)

[Hint: You will need to remember the power series for cos θ and sin θ given in Appendix B. You may wonder whether it is legitimate to regroup the terms of an infinite series as recommended here; for power series like those in this problem, it is legitimate.]