Signals and Systems - Electrical Engineering

0.4 Complex or Real? 23

0.4.2 Functions of a Complex Variable

Just like real-valued functions, functions of a complex variable can be defined. For instance, the

logarithm of a complex number can be written as

v=log(z)=log(|z|ejθ)=log(|z|)+jθ

by using the inverse connection between the exponential and the logarithmic functions. Of particular

interest in the theory of signals and systems is the exponential of complex variablezdefined as

v=ez=ex+jy=exejy

It is important to mention that complex variables as well as functions of complex variables are more

general than real variables and real-valued functions. The above definition of the logarithmic function

is valid whenz=x, withxa real value, and also whenz=jy, a purely imaginary value. Likewise, the

exponential function forz=xis a real-valued function.

Euler’s Identity

One of the most famous equations of all times^6 is

1 +ejπ= 1 +e−jπ= 0

due to one of the most prolific mathematicians of all times, Leonard Euler.^7 The above equation can

be easily understood by establishing Euler’s identity, which connects the complex exponential and

sinusoids:

ejθ=cos(θ)+jsin(θ) (0.18)

One way to verify this identity is to consider the polar representation of the complex number cos(θ)+

jsin(θ), which has a unit magnitude since

√

cos^2 (θ)+sin^2 (θ)=1 given the trigonometric identity

cos^2 (θ)+sin^2 (θ)=1. The angle of this complex number is

ψ=tan−^1

[

sin(θ)

cos(θ)

]

=θ

Thus, the complex number

cos(θ)+jsin(θ)= 1 ejθ

which is Euler’s identity. Now in the case whereθ=±π the identity implies thate±jπ=−1,

explaining the famous Euler’s equation.

(^6) A reader’s poll done byMathematical Intelligencernamed Euler’s identity the most beautiful equation in mathematics. Another poll by
Physics Worldin 2004 named Euler’s identity the greatest equation ever, together with Maxwell’s equations. Paul Nahin’s bookDr. Euler’s
Fabulous Formula(2006) is devoted to Euler’s identity. It states that the identity sets “the gold standard for mathematical beauty” [73].
(^7) Leonard Euler (1707–1783) was a Swiss mathematician and physicist, student of John Bernoulli, and advisor of Joseph Lagrange. We
owe Euler the notationf(x)for functions,efor the base of natural logs,i=
√
−1,πfor pi, 6 for sum, the finite difference notation 1 ,
and many more!