Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 4 ppt
Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 4 ppt
... approximation and the exact solution are plotted in Figure 48 .5. The value of the functional is
J = −
5
2 16
≈ −0.023 148 1.
0.2
0 .4 0 .6
0.8
1
0.01
0.02
0.03
0. 04
0.05
0. 06
0.07
Figure 48 .5: One Term ... is
y(x) = −
16
π(π
2
+ 4)
sin
πx
2
.
This one term approximation and the exact solution are plotted in Figure 48 .9. The value of the functional is
J = −
64
π
2
(...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 4 pptx
... −2x)
2
+ 2x(12 − 2x)(−2)
= 4( x 6)
2
+ 8x(x − 6)
= 12(x −2)(x − 6)
There are critical points at x = 2 and x = 6.
f
(x) = 12(x −2) + 12(x − 6) = 24( x 4)
Since f
(2) = 48 < 0, x = 2 is a ... of
4
3
√
π
cm and a height of
4
3
√
π
.
1 06
a
x x
x
x
x
x∆
1
2
3
i
n-2
n-1
b
f( )
ξ
1
Figure 4. 2: Divide -and- Conquer Strategy for Approximating a Definite Integral.
4. 2....
Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 4 ppt
... determine x and y in terms of ξ and ψ.
x = −
3
4
(ξ + ψ)
1/3
, y =
ψ −ξ
2
We calculate the derivatives of ξ and ψ.
ξ
x
=
√
−x =
3
4
(ξ + ψ)
1 /6
, ξ
y
= −1
ψ
x
=
3
4
(ξ + ψ)
1 /6
, ψ
y
= ... multiplicative constant.
17 16
36. 3 Exercises
Exercise 36. 1
Classify and transform the following equation into canonical form.
u
xx
+ (1 + y)
2
u
yy
= 0
Hint, Solution
Exerc...
Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 7 ppt
... + bˆg(s)
d
dt
f(t) s
ˆ
f(s) − f(0)
2 244
∞
n=1
1
n
5
= ζ(5)
∞
n=1
(−1)
n+1
n
5
=
15ζ(5)
16
∞
n=1
1
n
6
=
π
6
945
∞
n=1
(−1)
n+1
n
6
=
31π
6
30 240
2 240
... equation for z,
−
1
2
Uz
2
+
5
2
z
4
+ 5zP (z) −
1
2
1
4
(P
(z))
2
+
1
2
P
(z)P (z) = 0
4Uz
2
+ 20z
4
+ 40 zP (z) −(P
(z))
2
+ 4P
(z)P (z) = 0
Substituting P (z) = az
3
+ bz
2
yields
(20 + 40 a ....
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 2 ppt
... 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 2. 14: The ... components.
• If a · b = 0 then either a and b are orthogonal, (perpendicular), or one of a and b are zero.
26
1
2
1
2
2
4
6
8
10
1
2
Figure 1.11: Plots of f(x) = p(x)/q(x).
1 .6 H...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 3 pptx
... in 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 ... 3 .6
Note that the function
√
x + δ −
√
x is a decreasing function of x and an increasing function of δ for positive x and
δ. Bound this function for fixed δ.
Consider any...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 7 ppt
... + 4
63 − ı 16
=
3 + 4
63 − ı 16
63 + ı 16
63 + ı 16
= −
253
42 25
− ı
2 04
4225
215
2.
(1 − ı)
7
=
(1 − ı)
2
2
(1 − ı)
2
(1 − ı)
= (−ı2)
2
(−ı2)(1 − ı)
= ( 4) (−2 −ı2)
= 8 + ı8
Solution 6. 5
1.
z
z
=
x ... 2
−10
e
−ı10π/3
=
1
10 24
cos
−
10π
3
+ ı sin
−
10π
3
=
1
10 24
cos
4
3
− ı sin
4
3
=
1
10 24
−
1
2
+ ı
√
3
2
= −
1
2 048
+ ı
√
3
2 048...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 8 ppt
... cuts and define a branch so that z = 4 does not lie on a cut, and such that w = 6 when z = 4.
Hint, Solution
Exercise 7. 24
Give the number of branches and locations of the branch points for the ... m, n ∈ Z
z =
4m
1 + 4n
, m, n ∈ Z
There are an infinite set of rational numbers for which ı
z
has 1 as one of its values. For example,
ı
4/ 5
= 1
1/5
=
1,
e
ı2π/5
,
e
4 /5
,
e...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 9 ppt
... ı
1 /4
=
2
e
−ıπ /6
1 /4
=
4
√
2
e
−ıπ/ 24
1
1 /4
=
4
√
2
e
ı(πn/2−π/ 24)
, n = 0, 1, 2, 3
1
ı /4
=
e
(ı /4) log 1
=
e
(ı /4) (ı2πn)
=
e
−πn/2
, n ∈ Z
308
7.11 Hints
Cartesian and Modulus-Argument Form
Hint ... /2
(
e
ız
−
e
−ız
) /(ı2)
= 47
e
ız
+
e
−ız
= 47
e
ız
−
e
−ız
46
e
ı2z
48 = 0
ı2z = log
24
23
z = −
ı
2
log
24
23
z = −
ı
2
ln
24
23
+ ı2πn
, n...
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) .
44 7
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...
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