Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 8 docx
Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 8 docx
... equations, 8 06
regular Sturm-Liouville problems, 1420
properties of, 14 28
residuals
of series, 527
residue theorem, 63 1, 22 26
principal values, 64 3
residues, 62 7, 2225
of a pole of order n, 62 7, 22 26
Riccati ... 547
radius of convergence, 541
uniformly convergent, 539
principal argument, 1 86
principal branch, 6
principal root, 199
principal value, 63 4, 15 48
pure ima...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 8 ppt
... See
Figure 7. 18 and Figure 7.19 for plots of the real and imaginary parts of the cosine and sine, respectively. Figure 7.20
shows the modulus of the cosine and the sine.
The hyperbolic sine and cosine. ... y = 0.
This becomes the two equations (for the real and imaginary parts)
sin x cosh y = 0 and cos x sinh y = 0.
Since cosh is real-valued and positive for real argume...
Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 8 pot
... and expand in partial fractions.
Hint 13 .6
Hint 13.7
Cauchy Principal Value for Real Integrals
Hint 13 .8
68 0
f(a) =
2π
√
1 − a
2
Complex-Valued a. We note that the integral converges except for ... order
68 6
Hint, Solution
Exercise 13.23
By methods of contour integration find
∞
0
dx
x
2
+ 5x + 6
[ Recall the trick of considering
Γ
f(z) log z dz with a suitably chosen co...
Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 8 ppsx
... the form of a particular solution. This form will contain some unknown parameters. We substitute this form
into the differential equation to determine the parameters and thus determine a particular ... are y
1
=
e
x
, and y
2
=
e
−x
. The Green function has the form
G(x|ξ) =
c
1
e
x
+c
2
e
−x
for x < ξ
d
1
e
x
+d
2
e
−x
for x > ξ.
Since the solution must be bounded for all x,...
Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 1 docx
... q
N−j
a
j
(α
2
)
and
d
n
= lim
α→α
2
d
dα
(α − α
2
)a
n
(α)
for n ≥ 0.
12 08
0.2 0.4 0 .6 0 .8
1
1.2
1.4
0.7
0 .8
0.9
1.1
1.2
Figure 23.1: Plot of the Numerical Solution and the First Three ... and
a
n
= −
3
16n(n + 1)
a
n−1
for n ≥ 1.
This difference equation has the solution
a
n
= a
0
n
j=1
−
3
16j(j + 1)
= a
0
−
3
16
n
n
j=1
1
j(j + 1)
= a
0
−
3
16...
Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 4 docx
... think that this formula is just a
shade less than worthless. When solving an eigenvalue problem you have to find the eigenvalue s before you determine
the eigenfunctions. Thus this formula could ... not be used to compute the eigenvalues. However, we can often use the
formula to obtain information about the eigenvalues before we solve a problem.
Example 27.4.2 Consider the self-adjoint eigenva...
Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 7 docx
... the form
p
2
y
+ p
1
y
+ p
0
y = µy, for a ≤ x ≤ b,
α
1
y(a) + α
2
y
(a) = 0, β
1
y(b) + β
2
y
(b) = 0,
where the p
j
are real and continuous and p
2
> 0 on [a, b], and the α
j
and ... graph.
Hint 29 .6
Hint 29.7
Find the solution for λ = 0, λ < 0 and λ > 0. A problem is self-adjoint if it satisfies Green’s identity.
Hint 29 .8
Write the equation in self-adjoi...
Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 8 potx
... origin is
sin t = t −
t
3
6
+ O(t
5
).
Thus the Laplace transform of sin t has the behavior
L[sin t] ∼
1
s
2
−
1
s
4
+ O(s
6
) as s → +∞.
1 488
We use integration by parts and utilize the homogeneous ... have
1
ı2π
B
+
L
+
+
C
+
L
−
e
st
1
√
s
ds = 0.
14 86
Example 31.1.3 Consider the Laplace transform of the Heaviside function,
H(t − c) =
0 for t < c
1 for t &...
Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 9 docx
... equation.
sˆy
3
= −ˆy
3
−
ˆy
3
s
−
ˆy
3
s
2
+
6
s
4
(s
3
+ s
2
+ s + 1)ˆy
3
=
6
s
2
ˆy
3
=
6
s
2
(s
3
+ s
2
+ s + 1)
.
We solve for ˆy
1
.
ˆy
1
=
6
s
4
(s
3
+ s
2
+ s + 1)
ˆy
1
=
1
s
4
−
1
s
3
+
1
2(s ... ıa
∞
0
, for (s) > 0
=
1
ı2
1
s − ıa
−
1
s + ıa
L[sin(at)] =
a
s
2
+ a
2
for (s) > 0
1507
Since
ˆ
f
(s) is defined for (s) > 0,
ˆ
f(s) is analytic for (...
Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 2 docx
... 1
2t
+
z
2
4t
2
t
−n−1
e
t−z
2
/4t
dt
162 9
n Γ(n)
√
2πx
x−1/2
e
−x
relative error
5 24 23 .60 38 0.0 165
15 8. 71 783 · 10
10
8 .66 954 · 10
10
0.0055
25 6. 204 48 · 10
23
6. 183 84 · 10
23
0.0033
35 2.95233 · 10
38
2.94531 · 10
38
0.0024
45 ... A stands for either I or K.
164 9
1
2
3
4
2
4
6
8
10
Figure 34.4: Modified Bessel Functions
I
ν
and K
ν
are linearly independ...
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