... 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 use of ... 560
18.6.2 Orthogonality
of
Eigenfunctions 562
Transforms 559
MATHEMATICAL
METHODS
IN
SCIENCE
AND ENGINEERING
MATHEMATICS AND MIND
5
1.3
MATHEMATICS AND
MIND
Almost, everywhere ... discussion of infinite series: tests of
convergence, properties, power
series,
and uniform convergence along with
the methods
of
evaluating
sums
of
infinite series. An interesting section...
... MATHEMATICAL
METHODS
IN
SCIENCE
AND ENGINEERING
PREFACE
xviii
encountered special functions inscienceand engineering. This is also very
timely, because during the first year
of
graduate ... 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 use of ... discussion of infinite series: tests of
convergence, properties, power
series,
and uniform convergence along with
the methods
of
evaluating
sums
of
infinite series. An interesting section...
... relevant links of interest to readers.
S.
BAYIN
OD
TU
Ankam/TURKE
Y
April
2006
MATHEMATICAL
METHODS
IN
SCIENCE
AND ENGINEERING
Preface
Courses on mathematicalmethodsof 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 ...
Applications
of
Diflerintegrals inScience
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...
... Interpretation
of
V(x)
in
the Bloch Equation
20.6 Methods
of
Calculating Path Integrals
20.6.1 Method
of
Time Slices
20.6.2 Evaluating Path Integrals with the ESKC
20.6.3 Path Integrals ...
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 ... NA L DERIVATIVES and INTEGRA
LS:
“DIFFER INTEGR A LS”
14.1
Unified Expression
of
Derivatives and Integrals
14.1.1
Notation and Definitions
14.1.2
The nth Derivative
of
a Function
293...
... 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 ... 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 of books, ... 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...
... 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...
... majority of the second-order linear ordinary differential equations
of
sci-
ence andengineering can
be
conveniently expressed in terms
of
the three
parameters
(a,
b,
c)
of
the hypergeometric ... second kinds are
linearly independent, and they both satisfy the Chebyshev Equation
(5.33).
In terms of
x
the Chebyshev polynomials are written as
and
96
BESSEL
FUNCTIONS
thus obtaining ... and Hilbert. What
is
important in most applications
is
that any sufficiently well-behaved and
at
least piecewise continuous function
can be expressed
as
an infinite series in terms of...
... 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 mean
of
the ...
is
square integrable means that the
integral
of
the square of the derivative is finite for all the subintervals of the
fundamental domain
[a,
b]
in which the function is continuous.
8.5 ...
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
the...
... discussion of Cartesian coordinates and Cartesian tensors
to generalized coordinates and general tensors. These definitions can also
be
used for defining tensors in spacetime and also for tensors in ...
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 ...
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...
... subgroup of
U(n).
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 ... is called
pseudo-Euclidean.
In
this line element
c
is the speed
of
light representing the maximum velocity
in nature. An interesting property
of
the Minkowski spacetime is that two ...
time can be written in terms
of
basis vectors
as
Relative Orientation
of
Axes in
K
and
K
Frames
A
=
(Ao,A1,A2,A3)
(10.331)
(10.332)
In terms
of
another Minkowski frame, the...
... REPRESENTATIONS
OF
s
u(2)
(11.331)
From
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 ...
SPHERICAL HARMONICS AND REPRESENTATIONS
OF
R(3)
An elegant and also useful way
of
obtaining representations of
R(3)
is to con-
struct them through the transformation properties
of
the spherical ... values 1,2,3 and they correspond
to
x,
y,
z,
respectively.
SPHERICAL HARMONICS AND REPRESENTATIONS
OF
R(3)
261
11.11.10
To
find the inverse matrices we invert
Inverse
of
the
d;,,(p)...