MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 16 pptx
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 16 pptx
...
dk'g(k')eikfZ
VG
-"
and
Their inverse Fourier transforms are
and
Using these in Equation (19. 116) we get
which gives
us
Substituting this in Equation (19.118) we obtain
Writing
g(k') ...
Xj
and take its complex conjugate
as
(18.97)
Multiplying Equation (18.96) by
Xjy;(z)
and Equation (18.97) by
Xiyi(z),
and integrating over
x
in the interv...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 4 pptx
... equations in
physics and engineering 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 ... 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 lin...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 1 potx
... definition, 429
Cauchy integral formula, 390
Griinwald definition
differintegrals, 385
Laplace transforms, 396
484
Fractional derivatives
MATHEMATICAL
METHODS
IN
SCIENCE
AND ENGINEERING ...
11.15
Hilbert Space and Quantum Mechanics
11 .16
Continuous Groups and Symmetries
11 .16. 1
One-Parameter Point Groups and Their
11 .16. 2
Transformation of Generators and...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 2 pdf
... part
of the learning process.
In a vast area like mathematical methods in science and 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, w...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 3 docx
... 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...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 5 ppsx
...
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 ... defined as
Ladder Operators
for
the Spherical Harmonics
Using Equation (9 .162 )
we
write
Using
(9 .162 )
(9 .163 )
(9 .164 )
(9 .165 )
146
STURM-LIOUVILLE SYSTEMS AND THE F...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 6 pps
... 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 ...
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
Starti...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 7 pdf
...
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 ...
be expressed in terms of the poten-
tials
-A'
and
4
as
In the Lorentz gauge
2
and
4
satisfy
and
(10.373)
(10.374)
(10.375)
(10.376)
(10.377)
respectively. Defini...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 8 pps
... GROUP AND ITS LIE ALGEBRA
245
and
and introduce the parameters
(1
1 .166 )
8=
vf8:+8;+8,2
and
(11 .167 )
(1
1 .168 )
so
that we
can
summarize these results
as
L
=
X.68
+
v-pp
(11 .169 ) ...
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‘)
coordin...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 9 pps
... useful in finding solutions
of
Laplace equation in two dimensions.
2.
The method of analytic continuation
is
very useful in finding solutions
of differential equations and evaluating some ... complex techniques are very helpful in certain problems
of
physics and engineering, which are essentially problems defined in
the real domain, complex numbers in quantum mech...
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