Simple Nature - Light and Matter

Whenx= iφis imaginary, the quantity (1 +iφ/n) represents a

number lying just above 1 in the complex plane. For largen, (1 +

iφ/n) becomes very close to the unit circle, and its argument is the

small angleφ/n. Raising this number to the nth power multiplies

its argument byn, giving a number with an argument ofφ.

Euler’s formula is used frequently in physics and engineering.

Trig functions in terms of complex exponentials example 32

.Write the sine and cosine functions in terms of exponentials.

.Euler’s formula forx=−iφgives cosφ−isinφ, since cos(−θ) =

cosθ, and sin(−θ) =−sinθ.

cosx=

ei x+e−i x

2

sinx=

ei x−e−i x

2 i

A hard integral made easy example 33

.Evaluate ∫

excosxdx

.This seemingly impossible integral becomes easy if we rewrite

the cosine in terms of exponentials:

∫

excosxdx

=

∫

ex

(

ei x+e−i x

2

)

dx

=

1

2

∫

(e(1+i)x+e(1−i)x) dx

=

1

2

(

e(1+i)x

1 +i

+

e(1−i)x

1 −i

)

+c

Since this result is the integral of a real-valued function, we’d like

it to be real, and in fact it is, since the first and second terms are

complex conjugates of one another. If we wanted to, we could

use Euler’s theorem to convert it back to a manifestly real result.^5

10.5.7 Impedance

So far we have been thinking in terms of the free oscillations of a

circuit. This is like a mechanical oscillator that has been kicked but

then left to oscillate on its own without any external force to keep

the vibrations from dying out. Suppose an LRC circuit is driven

with a sinusoidally varying voltage, such as will occur when a radio

(^5) In general, the use of complex number techniques to do an integral could
result in a complex number, but that complex number would be a constant,
which could be subsumed within the usual constant of integration.
628 Chapter 10 Fields