Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 7 ppt
Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 7 ppt
... 0
e
−c|ω|
1
x − ıα
, α > 0
0 for ω > 0
ı
e
αω
for ω < 0
1
x − ıα
, α < 0
ı
e
αω
for ω > 0
0 for ω < 0
1
x
−
ı
2
sign(ω)
H(x − c) =
0 for x < c
1 for x > c
1
ı2πω
e
−ıcω
e
−cx
H(x), ... ζ(3)
∞
n=1
(−1)
n+1
n
3
=
3ζ(3)
4
∞
n=1
1
n
4
=
π
4
90
∞
n=1
(−1)
n+1
n
4
=
7
4
72 0
2239
Appendix I
Continuous Transforms
I.1 Properties of Laplace Transforms...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 7 ppt
... 105 + ı88
Solution 6. 4
1.
2 + ı
6 − (1 − ı2)
2
=
2 + ı
−1 + ı8
2
=
3 + ı4
63 − ı 16
=
3 + ı4
63 − ı 16
63 + ı 16
63 + ı 16
= −
253
4225
− ı
204
4225
215
2.
(1 − ı)
7
=
(1 − ı)
2
2
(1 ... real-variable
counterparts.
7. 1 Curves and Regions
In this section we introduce curves and regions in the complex plane. This material is necessary for the study of
branch poin...
Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 4 ppt
... is
y(x) = x(1 − x)
71
369
+
7
41
x
.
This two term approximation and the exact solution are plotted in Figure 48 .6. The value of the functional is
J = −
1 36
5535
≈ −0.024 570 9.
2. Exact Solution. ... solutions unless
L is greater than about 1.9 366 . If L is smaller than this, the rope would slip off the pin. For L = 2, c
1
has the values
0.4 265 and 0 .75 24. The larger va...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 2 ppt
... is the angle from a to b and n is a unit vector that is orthogonal to a and b and in the direction s uch that
the ordered triple of vectors a, b and n form a right-handed system.
29
a
b
b
θ
b
Figure ... points (1, 2, 3), (2, 3, 1) and (3, 1, 2)? What is the distance
from the point (2, 3, 5) to the plane?
Hint, Solution
37
x
z
yj
i
k
z
k
j
i
y
x
Figure 2 .7: Right and left handed...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 3 pptx
... Figure 3.13.
2
4
6
8
10
0.2
0.4
0 .6
0.8
1
Figure 3.13: Plot of x
20
/20!.
Note that the error is very small for x < 6, fairly small but non-negligible for x ≈ 7 and large for x > 8. The ... function of x and an increasing function of δ for positive x and
δ. Bound this function for fixed δ.
Consider any positive δ and . For what values of x is
1
x
−
1
x + δ
>...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 4 pptx
... 1] and −1 ≤ cos(x
0
) ≤ 1, the approximation
sin x ≈ x −
x
3
6
+
x
5
120
has a maximum error of
1
5040
≈ 0.000198. Using this polynomial to approximate sin(1),
1 −
1
3
6
+
1
5
120
≈ 0.84 166 7.
1 07
2.
f
(0) ... of x and an increasing
function of δ for positive x and δ. Thus for any fixed δ, the maximum value of
√
x + δ −
√
x is bounded by
√
δ.
Therefore on the interval (0,...
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. ... Solution
2 87
-2
-1
0
1
2
x
-2
-1
0
1
2
y
0
0.5
1
-2
-1
0
1
2
x
-2
-1
0
1
2
x
-2
-1
0
1
2
y
-2
0
2
-2
-1
0
1
2
x
Figure 7. 25: Plots of |z
1/2
| and Arg
z
1/2
.
Im...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 9 ppt
... angles from each of the branch points in the finite complex plane.
Hint 7. 32
Hint 7. 33
Hint 7. 34
Hint 7. 35
Hint 7. 36
Hint 7. 37
300
Now to define the branch. We make a choice of angles.
z + 3 = r
1
e
ıθ
1
, ... coordinates.
e
z
2
=
e
(x+ıy)
2
=
e
x
2
−y
2
+ı2xy
=
e
x
2
−y
2
304
25 50
75
100
-1
1
Figure 7. 43: The values of ı
ı
.
See Figure 7...
Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 2 pptx
... =
1
r
∂
∂r
r
∂u
∂r
+
1
r
2
∂
2
u
∂θ
2
= 0.
Therefore u is harmonic and is the real part of some analytic function.
Example 9.3.9 Find an analytic function f(z) whose real part is
u(r, θ) = r (log r cos θ −θ sin θ) .
4 47
The complex ... ı and an isolated essential singularity at z = 1.
(b)
sin(z − 3)
(z − 3)(z + ı)
6
has a removable singularity at z = 3, a pole of order 6 at...
Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 3 ppt
... dx
≤
b
a
|f(x)||dx| ≤ (b − a) max
a≤x≤b
|f(x)|.
466
with a, b and c complex-valued constants and d a real constant. Substituting z = x + ıy and expanding products
yields,
a
x
3
+ ı3x
2
y − 3xy
2
− ... the integrand, ıφ
x
− φ
y
is continuous and vanishes for all C in Ω, we conclude that the integrand is identically
zero. This implies that the Cauchy-Riemann equation,
φ...
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