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Mathematical Methods for Physicists: Supplement with Q&A
Paul Wenk
∗
Institut f¨ur Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany
(Dated: July 1, 2019)
Collection of frequently asked questions which came up during the course mathematical methods
for physicists at the University of Regensburg.
I. ANALYSIS
A. Partial Fraction Decomposition
The partial fraction decomposition is a way of expressing a rational function by a sum of fractions with a simpler
denominator. Let us assume f is a proper rational function, i.e., a rational function in which the degree of the poly-
nomial in the nominator is no greater than the degree of the polynomial in the denominator. Then the decompsition
can be written as follows
1
,
f(x) =
P (x)
Q(x)
=
M
X
i=1
j
i
X
k=1
A
ik
(x − a
i
)
k
+
N
X
i=1
k
i
X
k=1
B
ik
x + C
ik
(x
2
+ b
i
x + c
i
)
k
, x, a
i
, b
i
, c
i
∈ R . (1)
Here, the number of different real roots a
i
is M, indexed with i. Each root a
i
has a degeneracy j
i
. In general, also
complex root can appear. However, due to the complex conjugate root theorem the complex roots appear only in
complex conjugate pairs. Thus, to avoid the calculation with complex numbers one leaves quadratic polynomials
unfactorised, x
2
+ b
i
x + c
i
. The complex conjugate pairs can have also a degeneracy, here named by k
i
.
B. Dirac-Delta Distribution
Show that the following statement
Z
∞
−∞
dx f(x)δ(x) := lim
n→∞
Z
∞
−∞
dx f(x)δ
n
(x) = f(0) (2)
hold for a sequence of Gaussian distributions
δ
n
(x) =
n
√
π
e
−n
2
(x−x
0
)
2
(3)
Proof. We need the following preface:
(a) We evaluate
Z
∞
0
dx xe
−n
2
x
2
=
1
2n
2
Z
∞
0
dy e
−y
=
1
2n
2
(4)
where we used the substitution
y = (nx)
2
⇒ dy = 2xn
2
dx, dx =
1
2xn
2
dy. (5)
(b)
∀
n∈N
∗
:
Z
∞
−∞
dx δ
n
(x) = 1.
For the proof, see exercise sheet.
(c) The mean value theorem states that if f is a continuous function on a closed interval [a, b] there exists a ξ ∈ (a, b)
such that (see Fig. 1)
f
0
(ξ) =
f(b) − f(a)
b − a
. (6)