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Mathematical Series
Mathematical series representations are very useful tools for describing images or for solving/approximating the solutions to imaging problems. The may be used to “expand” a function into terms that are individual monomial expressions (i.e., “powers”) of the coordinate
Geometric Series
Adjacent terms in a geometric series exhibit a constant ratio, e.g., if the scale factor for adjacent terms in the series is t, the series has the form:
1 + t + t^2 + t^3 + · · · =
X^ ∞
n=
tn
If |t| < 1 , this solution converges to a simple (and EASILY remembered) expression:
X^ ∞
n=
tn^ = 1 1 − t
if |t| < 1
This series pops up frequently in science and it is useful to remember the solution. We may “turn the problem around” by using a truncated series as an approximation for the ratio:
1 1 − t
X^ ∞
n=
tn^ ∼=
X^ N
n=
tn
where N is some maximum power in the series
Examples:
= (0.1)^0 + (0.1)^1 + (0.1)^2 + (0.1)^3 + · · ·
This series converges fairly quickly and because of the small value of t; this series may be truncated after a few terms and still obtain a fairly accurate value.
1
- 25
= (0.75)^0 + (0.75)^1 + (0.75)^2 + (0.75)^3 + (0.75)^4 + (0.75)^5 + · · ·
Note that this series converges slowly because t is relatively “large;” the sum of a few terms is a poor approximation to the end result.
Finite Geometric Series
The truncated geometric series also may be rewritten into a simple expression. Consider the finite series that inclues N + 1 terms:
X^ N
n=
tn^ = 1 + t + t^2 + t^3 + · · · + tN
X^ ∞
n=
tn^ −
X^ ∞
n=N +
tn
We may write this as the difference of two infinite geometric series:
X^ N
n=
tn^ =
1 + t + t^2 + t^3 + · · · tN^ + tN^ +1^ + · · ·
tN^ +1^ + tN+2^ + · · ·
X^ ∞
n=
tn^ −
X^ ∞
n=N+
tn
Now change the summation variable for the second infinite series from n to u ≡ n − (N + 1) =⇒ n = u + N + 1: (^) ∞ X
n=N +
tn^ =
X^ ∞
u=
tu+(N^ +1)^ =
X^ ∞
u=
tutN^ +1^ = tN+1^ ·
X^ ∞
u=
tu^ = tN^ +1^ · 1 1 − t
The expressions for the two series may now be combined:
X^ N
n=
tn^ =
X^ ∞
n=
tn^ −
X^ ∞
n=N
tn
μ 1 1 − t
− tN+1^ ·
1 − t X^ N
n=
tn^ =
1 − tN^ + 1 − t
if |t| < 1
This is also often shown in the form where the maximum power in the series is N − 1 so that there are N terms: NX− 1
n=
tn^ =
1 − tN 1 − t
if |t| < 1
Examples:
X^4
n=
(0.1)n^ = 1 + 0.1 + 0.01 + 0.001 + 0.0001 = 1. 1111
t = 0. 1 , N = 4 =⇒ 1 −^ t
N + 1 − t
=^1 −^ (0.1)
5 1 − (0.1)
X^3
n=
(0.75)n^ = 2. 7344
1 − tN+ 1 − t
=^1 −^ (0.75)
4 1 − (0.75)
- Cube root
(1 − x)
(^13) = 3
p (1 − x) = 1 +
3 ·^ (−x) +
3
2 ·^ (−x)
3
6 ·^ (−x)
3 ·^ x^ −^
9 ·^ x
2 +^5
81 ·^ x
∼= 1 +^1
3 ·^ x^ −^
9 ·^ x
2 ∼= 1 +^1
3 ·^ x
μ 15 16
μ 1 − 1 16
∼= 1 +^1
μ − 1 16
=^47
μ 1 −
∼= 1 +^1
μ −
μ 1 16
μ 1 −
∼= 1 +^1
μ −
μ −
μ −
Exponential Series
Without proof, we can write e raised to a numerical power u as a series in the powers of u:
exp [u] = eu^ =
X^ ∞
n=
un n!
=^1 0!
2 2!
3 3!
= 1 + u +^1 2
u^2 +^1 6
u^3 + · · ·
which may be easily generalized to any base:
au^ = exp [u · log [a]] =
X^ ∞
n=
[u · log [a]]n n!
e.g.
10 u^ =
X^ ∞
n=
[u · log [10]]n n!
∼=
X^ ∞
n=
(log [10])n n!
· un
where log [10] ∼= 2. 302585
Complex Exponential Series
The generalization of the exponential series for complex-valued powers:
exp [±iθ] = e±iθ^ =
X^ ∞
n=
(±iθ)n n!
=^1 0!
± (iθ) 1!
2 2!
∓ (iθ)
3 3!
4 4!
= 1 ± iθ +^1 2
i^2 θ^2 ∓ 1 6
i^3 θ^3 +^1 24
i^4 θ^4 ± 1 120
i^5 θ^5 + · · ·
μ 1 −
θ^2 2
θ^4 24
± i ·
μ θ −
θ^3 6
θ^5 120
From Euler relation: exp [+iθ] = cos [θ] + i sin [θ]
Equate real and imaginary parts:
cos [θ] =
0! −^
θ^2 2! +^
θ^4 4! −^ · · ·^ = 1^ −^
θ^2 2 +^
θ^4 24 −^ · · · θ^ lim→ 0 cos [θ] = 1
sin [θ] = θ −
θ^3 3!
θ^5 5!
− · · · = θ −
θ^3 6
θ^5 120
lim θ→ 0
sin [θ] = θ
