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periodicity of exponential function

period of exponential function
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32A05 no label found30D20 no label found


I am finding empirically that exp(2 Pi i b) - exp(2 Pi i a) = 0 when b-a is an integer. I would like to see a proof or discussion of why this is so.

exp(2Pi i b) - exp(2Pi i a)

= cos(2Pi b) + i sin(2Pi b) - (cos(2Pi a) + i sin(2 Pi a)) (by Euler's formula)

= cos(2Pi b) - cos(2 Pi a) + sin(2Pi b) - sin(2 Pi a) (by rearranging)

= - 2 sin((2Pi b + 2Pi a)/2) sin((2Pi b-2Pi a)/2) + 2i (sin((2Pi b - 2Pi a)/2) cos((2Pi b + 2Pi a)/2) ) (by trig identities **)

= -2 sin(Pi (b + a)) sin(Pi (b - a)) + 2i sin(Pi (b-a)) cos(Pi (b-a)) (by cancelling and rearranging)

Note that sin(Pi (b - a)) = 0 if b - a is an integer, so the right hand side goes to zero.

** sum to product identities

cos(b) - cos(a) = -2 sin((b+a)/2) sin((b-a)/2)

sin(b) - sin(a) = 2 sin((b-a)/2) cos((b+a)/2)

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