My title - Departamento de Matemática da Universidade do Minho

6 FLOWS 44
easily verify that indeed exp ′ = exp. The initial condition exp(0) = 1 is obvious. From absolute
convergence of the series and algebraic manipulation we also get the group property
exp(z + w) = exp(z) · exp(w)
for any z, w ∈ C, saying that exp is a homomorphism of the additive group C into the multiplicative
group C × = C\{0}. In particular, exp(−z) = 1/ exp(z), so that the exponential exp(z) is never 0.
This also justifies our notation exp(z) = e z , where
e = exp(1) = 1 + 1 1! + 1 2! + 1 3! + · · · ≃ 2.7182818284590452353602874713526624977572 . . .
(another famous irrational, actually a transcendental number!). For real time z = t, we recover
the familiar model of “exponential growth” t ↦→ e t , a strictly increasing function from the additive
group R onto the multiplicative group R + =]0, ∞[, growing faster than any power t n as t → ∞.
For pure imaginary times, say z = iθ, we get the Euler’s formula
)
)
e iθ =
(1 − θ2
2! + θ4
4! − . . . + i
(θ − θ3
3! + θ5
5! − . . . = cos(θ) + i sin(θ)
So, θ ↦→ e iθ defines a periodic function with period 2π, sending the real line R onto the unit circle
S = {z ∈ C s.t. |z| = 1}. Finally, the exponential exp is a periodic entire function with period i2π
which only omits the value 0, a holomorphic bijection of the cylinder C/i2πZ onto C\{0}.
Interest rates and the exponential. Let x be the annual interest payed for a deposit (so that
an interest of 0.2% mean x = 0.02). If the interest is payed once each year, an initial deposit of a
euros increases to
a + xa = a · (1 + x)
after one year. If, however, the interest is “computed” every six months, the same initial deposit
produces
a + x (
2 a + a + x ) x
(
2 a 2 = a · 1 + x ) 2
2
after one year. By induction, we see that if the interest is computed every 12/n months, after one
year we get a final capital of
(
a · 1 + x ) n
n
The limit of the gain factor as n → ∞,
(
E(x) = lim 1 + x ) n
n→∞ n
is another definition of the exponential function.
Population dynamics. The exponential models the dynamics of a population in a unlimited
environment. The Malthusian/exponential model 11 is
Ṅ = λN
where N(t) is the population at time t, and λ > 0 is some growth constant (the difference α − β
between the natality rate and the mortality rate). The solution is N(t) = N(0)e λt . If we retire at
fixed rate α > 0
Ṅ = λN − α
we have a non-trivial stationary solution N = α/λ, and the difference x(t) = N(t) − N is still
exponential.
This behaviour has to be compared with the super-exponential model
Ṅ = αN 2 .
11 T.R. Malthus, An Essay on the Principle of Population, London, 1798.