MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 11 ppt
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 1 potx
... Relativity
16 3
1
63
16 6
16 6
1
70
1
70
1
72
1
74
17 5
17 7
1
78
1
78
17 9
18 0
18 1
18 3
16 4
18 6
18 8
18 8
18 8
18 9
18 9
18 9
18 9
19 0
19 3
19 5
19 6
19 7
19 7
19 7
19 9
8 01
2 01
...
Hermitian Operators
in
Quantum Mechanics
Problems
10 7
10 7
I
08
11 0
11 0
11 1
11 1
11 2
11 3
11 4
11 5
11 8
9
STURM-LIOUVILLE SYST...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 2 pdf
... substitutions
kin
and
24 2-n
(2. 74)
to write
(2. 75)
n=O
C21 (2
-
n)!n!
(1
-
2n)!
Comparing this with the right-hand side of Equation
(2. 65),
which is
we obtain
(2. 76)
(2. 77)
2. 3.3
Recursion ... write
&s+l
=
-
(4s
+
3,
-
p2s
(O)
U2S +2(
avo),
s
=
0,1 ,2,
2 (2s +2)
(2. 1 32)
Substituting these in Equation
(2. 127 )
we finally obtai...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 3 docx
... 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
(3. 42). ... defined by setting
a0
=
1
in Equation
(3. 23)
as
3. 2
OTHER DEFINITIONS OF LAGUERRE POLYNOMIALS
3. 2.1
The generating function
of
the Laguerre polynomials is defined
as
Gener...
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 ... following integral definitions:
(6 .48 )
and
1
2
(1
-
t2)n-6
cosztdt,
(n
>
).
(6 .49 )
Jn(IL.1
=
6 .4
RECURSION RELATIONS
OF
THE BESSEL FUNCTIONS
Using the s...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 5 ppsx
...
+
.
d
sin+
+
p)
’
(9.89)
where
p
and
d
are integration constants.
With these
Ico
and
Icl
functions in Equation (9. 75) and the
p(m)
given
in Equation (9. 85) we obtain
r(z,m) ... 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)...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 6 pps
... use the
approximations
sin
6$
N
S+,
sin
6Cp
N
6Cp
(10. 96)
cos6$
N
1, cos6Cp
_N
1
to find
10
R1R2
=
[
6+
1
=R2R,.
(10.97)
6$
-64
Note that in terms of the generators ... 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...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 7 pdf
...
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. Defining
a
four-potential ... respectively
as
and
a;
=
(10. 275 )
.
(10. 276 )
For
the general linear transformation
[Eq.
(10.250)] matrix elements
a$
can
be obtained by using
dZa
dx...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 8 pps
... LORENTZ GROUP AND ITS LIE ALGEBRA
245
and
and introduce the parameters
(1
1.166)
8=
vf8: +8; +8, 2
and
(11.167)
(1
1.1 68)
so
that we
can
summarize these results
as
L
=
X. 68
+
v-pp ...
in
Quantum Mechanics
+
L=?;’xT,
(11. 188 )
by replacing position and momentum with their operator counterparts, that
is,
?+?,
as
L
=
4i-P
x
a‘.
(11. 189 )
(1 1,190...
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...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 10 pps
...
INTEGRALS
Many
of
the definite integrals encountered in physics and engineering can
be
evaluated by using the complex integral theorems and analytic continuation:
I.
Integrals
of
the
form ...
0
and
a-lrnl
#
0,
then
is called a singular point of order
m.
Definition I11
Essential singular point:
If
m
is infinity, then
a
is called an essential
singular point....
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 11 ppt
...
DERIVATIVES AND INTEGRALS
381
situation on the applied side
of
this branch of mathematics is now changing
rapidly, and there are now
a
growing number of research areas in science and
engineering ... techniques in
evaluating definite integrals and finding
sums
of
infinite
series.
In this chapter,
we introduce some of the basic properties
of
fractional calculus...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 12 doc
...
DIFFERINTEGRALS IN SCIENCE AND ENGINEERING
427
Fig.
14.7
Probability distribution in random walk and
CTRW
An important area of application for fractional derivatives is that the ex-
traordinary ...
dPaex
r*(a,z)
=
r(a)ecX-
dx-
14.7
APPLICATIONS OF DIFFERINTEGRALS
IN
SCIENCE AND
ENGINEERING
14.7.1
Continuous
Time
Random Walk (CTRW)
We have seen that the...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 13 pdf
...
INFINITE
PRODUCTS
471
Integrating Equation
(15.223)
gives
and finally the general expression
is
obtained
as
(15.225)
Applying this formula with
z
=
x to the sine and cosine functions ...
(z)
are
continuous functions, cannot
be
uniformly convergent in
any interval containing
a
discontinuity of
f
(x).
INFINITE
PRODUCTS
469
exists, then we say the infinite pro...
MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 14 doc
... for the above
integral
is
an infinite straight line passing through the point
y
and parallel
to the imaginary axis in the complex s-plane.
y
is chosen such that all the
singularities of ...
defined
as
and it
is
usually encountered in potential energy calculations in cylindrical
coordinates. Another useful integral transform
is
the Mellin transform:
(16 .14)
The Mell...
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...
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