Exponentials
The exponential of a real number x, written e^{ x} or exp(x), is defined by an infinite series,Also recall the infinite series expansion for cos and sin:
sin(θ) = θ - (θ^{ 3} / 3!) + (θ^{ 5} / 5!) + ...
The exponential of a complex number z is written e^{ z} or exp(z), and is defined in the same way as the exponential of a real number,
Note that e^{ 0} = 1, as expected.
The exponential of an imaginary number iθ is very interesting,
e^{ i}θ | = | 1 + (iθ) + ((iθ)^{ 2} / 2!) + ((iθ)^{ 3} / 3!) + ... |
= | [1 - (θ^{ 2} / 2) + (θ^{ 4} / 4!) - ...] + i[θ - (θ^{ 3} / 3!) + (θ^{ 5} / 5!) + ...] | |
= | cos(θ) + i sin(θ) |
This identity is known as Euler's formula:
Euler's formula is used heavily in this class in the analysis of the linear time invariant systems. It allows sinusoidal functions to be given as sums or differences of exponential functions,
sin(θ) = (e^{ i}θ - e^{ -i}θ) / (2i).
This proves useful because exponential functions turn out to be simpler mathematically (despite being complex valued) than sinusoidal functions.
An important property of the exponential function is the product formula:
Trigonometric Identities
We can obtain many trigonometric identities by combining Euler's formula and the product formula. For example, sinceand
we have the identity
Here is another example. Using
we get
cos(α + β) + i sin(α + β) | = | [cos(α) + i sin(α)] [cos(β) + i sin(β)] |
= | [cos(α) cos(β) - sin(α) sin(β)] | |
+ i [sin(α) cos(β) + cos(α) sin(β)]. |
Since the real part of the left side must equal the real part of the right side, we get the identity,
And, likewise with the imaginary parts,
It is much easier to remember Euler's formula and the product formula than these identities.