Q1. Show that the two definitions of asymptotic series, as given in class, are indeed equivalent. Solution:
Question:
Q1.
Show that the two definitions of asymptotic series, as given in class, are indeed equivalent.
Solution:
The two definitions given in class are:
Def #1: Given an asymptotic sequence {n=0∞ as z. We can say that f(z) as z if N
f(z) as z.
Def #2: for such N = 0, 1, 2,… f(z) , m z.
We now show that the two definitions are equivalent.
We have the asymptotic sequence {n=0∞ as z.
f(z) as z if N
f(z) as z.
This implies if we subtract the series of (N + 1) number of terms from f(z), we see that the Nth term of asymptotic sequence is much greater than f(z). For each value of z, a convergent series has a unique limiting sum but it does not provide any information on how well approximates for fixed n or on how it converges.
f(z)
- f(z) for m z. And N = 0, 1, 2,….
Clearly one definition imply others.
So, two definitions are equivalent.
Q2.
Based upon these definitions, derive the recursive equation for the coefficients, which was given in class.
Solution:
From definition we have,
Put m = 1
=
Put m = 2
=
= 0
Put m = 3
0 as
So,
Q3.
Obtain an asymptotic series for erfz = as zHint: recall that erf() = 1. Does the series converge? How many terms are needed to obtain a 10-5 accuracy for z = 3?
Solution:
erfz =
=
= 0z
=
=
We have erf(
This series converges for all z can be seen by:
| for any real z as n
For z = 3.
erf(3) = (That almost equal to
nearly, more than 30 terms are needed to get accuracy.
Q4.
Obtain 2-term approximations for the two large roots of:
.
Solution:
Let f(z) =
f(-z) =
f(z) has no change of sign in the expression. So, there are no positive real roots in f(z) = 0.
f(-z) has one change of sign in the expression. So, there are one negative real root in f(z) = 0.
So, other two roots are imaginary.
So, we cannot have two large roots as one root is negative and other two roots are imaginary.
College Physics
ISBN: 978-0495113690
7th Edition
Authors: Raymond A. Serway, Jerry S. Faughn, Chris Vuille, Charles A. Bennett