Show me the steps of determine red and it complete
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Show me the steps of determine red and it complete
![it can be very useful to compare their asymptotic behaviors. To do so, rewrite
equation (8.83) as
T(k+r)
I(k)
Yk
2 Re
(8.86)
where r = ri =
r*, and let C =
|C|e", where |C| is the magnitude of the
complex constant C and 0 is its constant phase. A property of the gamma
function is that for large k
T(k +r)
T(k)
k".
k-large
(8.87)
Therefore,
Yk
k-large
• 2Re{\C\e®k«+ib},
(8.88)
where
3
1
ib = r = -+
2
(8.89)
2
and
Yk
k-large
+[2|C\k"]Re{e'(b\nk+0)}
k° {A cos(b ln k) +B sin(b ln k)}
[(4).
kl/2
A cos
In k + B sin
In k
(8.90)
where A and B are two real arbitrary constants. As a comparison of y(x),
equation (8.81), and yk, equation (8.90) shows that they have exactly the
same asymptotic behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06b6425a-ab8a-4368-90d9-64663800e88c%2Fc74e1257-37e2-452b-936d-3f24a70b194b%2Fk5bndl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:it can be very useful to compare their asymptotic behaviors. To do so, rewrite
equation (8.83) as
T(k+r)
I(k)
Yk
2 Re
(8.86)
where r = ri =
r*, and let C =
|C|e", where |C| is the magnitude of the
complex constant C and 0 is its constant phase. A property of the gamma
function is that for large k
T(k +r)
T(k)
k".
k-large
(8.87)
Therefore,
Yk
k-large
• 2Re{\C\e®k«+ib},
(8.88)
where
3
1
ib = r = -+
2
(8.89)
2
and
Yk
k-large
+[2|C\k"]Re{e'(b\nk+0)}
k° {A cos(b ln k) +B sin(b ln k)}
[(4).
kl/2
A cos
In k + B sin
In k
(8.90)
where A and B are two real arbitrary constants. As a comparison of y(x),
equation (8.81), and yk, equation (8.90) shows that they have exactly the
same asymptotic behavior.
![8.3.2
Example B
Consider the following Cauchy-Euler differential equation
d²y
+ y = 0.
dx2
(8.77)
Its characteristic equation is
r(r – 1) +1= r2 – r +1= 0,
(8.78)
-
with solutions
1
= ra
i,
= V-1.
(8.79)
ri =
+
2
To obtain the general solution, the quantity x"1 must be calculated. This is
done as follows
(a) exp
V3
x"1
In x
(8.80)
= x
= X
• x
Therefore, y(x) is given by the expression
y(x) = x
A cos
In x + B sin
In x
(8.81)
Note that for x > 0, the solution oscillates with increasing amplitude.
The corresponding discrete version of equation (8.77) is
k(k + 1)A²yk + Yk
0,
(8.82)
and its characteristic equation is that given in equation (8.78). Therefore, the
general solution is
,r(k+r)
I(k)
T(k +r*)
+ C*
T(k)
Yk =
(8.83)
where r = r1 and C is an arbitrary complex number. Observe that the manner
in which the right side of equation (8.83) is written insures real values for yk.
This depends also on the fact that the gamma function T(z) is real-valued,
i.e., for z = x+ iy, I'(z*) = [T'(2)]*. The integral representation of the gamma
function allows this to be easily demonstrated, i.e.,
I(2) = | e-t-1dt.
(8.84)
Jo
Finally, while it is not to be expected that y(x), equation (8.81), and yk,
equation (8.83) have "exactly" the same mathematical structure for all x and
k, where the correlation between these variables is
x → Xk =
(Ax)k, Ax = 1,
(8.85)
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06b6425a-ab8a-4368-90d9-64663800e88c%2Fc74e1257-37e2-452b-936d-3f24a70b194b%2F1ce58zi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:8.3.2
Example B
Consider the following Cauchy-Euler differential equation
d²y
+ y = 0.
dx2
(8.77)
Its characteristic equation is
r(r – 1) +1= r2 – r +1= 0,
(8.78)
-
with solutions
1
= ra
i,
= V-1.
(8.79)
ri =
+
2
To obtain the general solution, the quantity x"1 must be calculated. This is
done as follows
(a) exp
V3
x"1
In x
(8.80)
= x
= X
• x
Therefore, y(x) is given by the expression
y(x) = x
A cos
In x + B sin
In x
(8.81)
Note that for x > 0, the solution oscillates with increasing amplitude.
The corresponding discrete version of equation (8.77) is
k(k + 1)A²yk + Yk
0,
(8.82)
and its characteristic equation is that given in equation (8.78). Therefore, the
general solution is
,r(k+r)
I(k)
T(k +r*)
+ C*
T(k)
Yk =
(8.83)
where r = r1 and C is an arbitrary complex number. Observe that the manner
in which the right side of equation (8.83) is written insures real values for yk.
This depends also on the fact that the gamma function T(z) is real-valued,
i.e., for z = x+ iy, I'(z*) = [T'(2)]*. The integral representation of the gamma
function allows this to be easily demonstrated, i.e.,
I(2) = | e-t-1dt.
(8.84)
Jo
Finally, while it is not to be expected that y(x), equation (8.81), and yk,
equation (8.83) have "exactly" the same mathematical structure for all x and
k, where the correlation between these variables is
x → Xk =
(Ax)k, Ax = 1,
(8.85)
%3D
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