Derive Euler's formula from the power series of the exponential

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Foundations of mathematical language

Mathematical Methods for Physics, Ch Complex numbers, the complex exponential and phasors

Problem

Derive Euler's formula

e^{i\theta} = \cos\theta + i\sin\theta

from the power series of the exponential, extended to an imaginary argument

i\theta

, by separating the terms with an even power of

i

from those with an odd power of

i

. ((a)) Sum the first four terms (

n = 0

3

) of the series for

e^{i\theta}

\theta = \pi/6\,\mathrm{rad}

, and compare the real and imaginary parts of the partial sum with

\cos(\pi/6)

and

\sin(\pi/6)

. ((b)) Using the exponential forms of

\sin\theta

and

\cos\theta

that follow from Euler's formula, prove that

\sin A\cos B = \tfrac12\left[\sin(A+B) + \sin(A-B)\right]

for any real

A

and

B

, and check it numerically at

A = 1.0\,\mathrm{rad}

B = 0.4\,\mathrm{rad}

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