... MATHEMATICAL
METHODS
IN
SCIENCE
AND ENGINEERING
MATHEMATICS AND MIND
5
1.3
MATHEMATICS AND
MIND
Almost, everywhere mathematics is
a
very useful and powerful language in
expressing ... part
of the learning process.
In a vast area like mathematicalmethodsinscienceand engineering, there
is
always room for new approaches,
new
applications, and new topics. In fact,
the number ... naturally comes from the inner
efficiency of our brain. Research on subjects like brain stimulators, hard
wiring of our brain, and mind reading machines are all aiming at
a
faster and
much more efficient...
...
(2.27)
MATHEMATICAL
METHODS
IN
SCIENCE
AND ENGINEERING
PREFACE
xviii
encountered special functions inscienceand engineering. This is also very
timely, because during the first year
of ... part
of the learning process.
In a vast area like mathematicalmethodsinscienceand engineering, there
is
always room for new approaches,
new
applications, and new topics. In fact,
the number ... naturally comes from the inner
efficiency of our brain. Research on subjects like brain stimulators, hard
wiring of our brain, and mind reading machines are all aiming at
a
faster and
much more efficient...
... relevant links of interest to readers.
S.
BAYIN
OD
TU
Ankam/TURKE
Y
April
2006
MATHEMATICAL
METHODS
IN
SCIENCE
AND ENGINEERING
Preface
Courses on mathematicalmethods of physics ... in physics, which
are
also offered by most engineering
departments. Considering that the audience in these coumes comes from all
subdisciplines
of
physics and engineering, the content and ... the growing in-
terest in interdisciplinary studies has brought scientists together from physics,
chemistry, biology, economy, and finance and has increased the demand for
these courses in which...
... definition, 429
Cauchy integral formula, 390
Griinwald definition
differintegrals, 385
Laplace transforms, 396
484
Fractional derivatives
MATHEMATICAL
METHODS
IN
SCIENCE
AND ENGINEERING ...
and
Engineering
424
14.7.2
Fractional Fokker-Planck Equations
427
Problems
429
14.7.1 Continuous Time Random Walk (CTRW)
424
15 INFINITE SERIES 431
15.1 Convergence
of
Infinite ...
Mathematical Methods: for Students of Physics and Related
Fields,
Springer Verlag, 2000.
Hassani,
S.,
Mathematical Physics,
Springer Verlag, second edition, 2002.
Hildebrand, F.B.,
Methods...
... part
of the learning process.
In a vast area like mathematicalmethodsinscienceand engineering, there
is
always room for new approaches,
new
applications, and new topics. In fact,
the number ... write
x
=
fl
in the generating function
Equation
(2.65)
we find
(2.85)
Expanding the left-hand side by using the binomial formula and comparing
equal powers oft, we obtain
9
(1)
=
1 ... and
for
meticulously reading Chapters
1
and
9
with
14
and
20.
I
also
thank Prof.
N.
K.
Pak
for many interesting and stimulating discussions, encouragement, and
critical reading...
... that the value of the integral in Equation
(3.38)
can
be
obtained
by expanding
in powers
of
t
and
s
and then by comparing the equal powers
of
tnsm
with
the left-hand side of Equation ...
(4.52)
'I
A
This Page Intentionally Left Blank
ORTHOGONALiTY
OF
LAGUERRE POLYNOMlALS
49
Interchanging the integral and the summation signs and integrating with
re-
spect
to
x
gives ... are very useful in cosmology and quantum
field
theory in
curved backgrounds. Both the spherical harmonics and the Gegenbauer poly-
nomials are combinations of sines and cosines. Chebyshev...
... second
kind. Note that
N,(z)
and
Jm(z)
are linearly independent even
for
the
integer values
of
m.
Hence it is common practice to take
N,(z)
and
Jm(z)
as
the two linearly independent ...
can obtain the following
recursion relations
and
First by adding and then by subtracting these equations we also obtain the
relations
m
Jm-l(X)
=
-Jm(IL.)
+
J:,(X)
(6.52)
X
and
(6.53) ... equations in
physics andengineering can be solved by the method of separation
of
variables.
This method helps us to reduce a second-order partial differential equation
into
a
set of ordinary...
... exists
a
minimum value,
mmin,
thus determining
X
as
X
=
mmin
(9.143)
To
find
mmin
we equate the two expressions
[Eqs.
(9.141)
and
(9.143)]
for
X
to obtain
(9.144)
mmin
=
-1. ...
Cancelling
m
on both sides and noting that
(9.166)
(9.167)
and using Equation (9.164) we finally write
Similarly
We now define the ladder operators
L+
and
L-
for
the
m
index
of ...
m=O
which converges absolutely and uniformly in all subintervals free of
points
of
discontinuity. At the points of discontinuity this series rep
resents
(as
in the Fourier series) the arithmetic...
...
axn
and
Adding the first two equations and subtracting the last one from the result
and after some rearrangement
of
indices we obtain
(10.197)
I0
COORDINATES
and
TENSORS
Starting with ... choosing three mutually orthogonal straight lines.
A
point is
defined by giving its coordinates,
(q,z2,q),
or by using the position vector
163
188
COORDINATES AND TENSORS
Hence
aa
and ...
1
0
-
sin
$
cosCp sin4cos$
cosdsin+ -sin4 cos~cos$
(10.93)
Reversing the order we get
cos
$
(10.94)
sin
$
sin
Cp
-
sin
$
cos
Cp
R2R1
=
0
cos
Cp
sin
4
[
sin
$
-
cos...
... the four-potential
is
defined
as
A"
=
(4,T)
and
224
CONTINUOUS GROUPS AND REPRESENTATIONS
inhomogeneous Lorentz group and introduce its Lie algebra. An advanced
treatment of spherical ...
Groups with infinite number of elements like
R(n)
are called
infinite groups.
A
group with finite number of elements is
called a
finite group,
where the number of elements in a finite group ...
+ae
d
(TaB)
r2sinOaW
+a
(WE)]
After simplifying, the Laplacian
is
obtained as
r2sin6
dr
34
r2sin28
a4
'
(10.239)
Here we have obtained
a
well-known formula in
a
rather straightfor-...
...
properties that remain invariant under similarity transformations. One such
254
CONTINUOUS GROUPS AND REPRESENTATIONS
Y’
Fig.
11.6
(z,y,z)
and the
(z‘,y’,z‘)
coordinates
we obtain (Fig. 11.6) ... 240
CONTINUOUS GROUPS AND REPRESENTATIONS
induces the following change in
a
function
f(v1,
~2):
-
f(V)
=
f
Mff,
P,
Y)VI'
(1 1.126)
Taking
the
active view we define an operator ...
the physical point
of
view a very important part of the group theory is
representing each element
of
the group with
a
linear transformation acting in
a
vector space. We now introduce the...