Euler's Formula: You can use the following process to justify Euler's formula
ei( = cos ( + i sin (
(a) Write out explicitly the first dozen or so terms of the Maclaurin series (the Taylor expansion about the origin) given by

(b) The series is valid for both real and complex numbers. Replace x by i( and write the expression for ei(.
(c) Simplify the results by using the periodicity of powers of i :
i0 = i4 = i8 = . . . = 1,
i1 = i5 = i9 = . . . = i,
i2 = i6 = i10 = . . . = -1,
i3 = i7 = i11 · · · = -i.
(d) Collect the real and imaginary terms.
(e) Obtain Euler's formula by recognizing the two Maclaurin series that appear in part (d).

Transcribed Image Text:

et n=0