Basic Technical Mathematics
11th Edition
ISBN: 9780134437705
Author: Washington
Publisher: PEARSON
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Chapter 25.2, Problem 56E
To determine
The expression for
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(b) In various places in this module, data on the silver content of coins
minted in the reign of the twelfth-century Byzantine king Manuel I
Comnenus have been considered. The full dataset is in the Minitab file
coins.mwx. The dataset includes, among others, the values of the
silver content of nine coins from the first coinage (variable Coin1) and
seven from the fourth coinage (variable Coin4) which was produced a
number of years later. (For the purposes of this question, you can
ignore the variables Coin2 and Coin3.) In particular, in Activity 8 and
Exercise 2 of Computer Book B, it was argued that the silver contents
in both the first and the fourth coinages can be assumed to be normally
distributed. The question of interest is whether there were differences in
the silver content of coins minted early and late in Manuel’s reign. You
are about to investigate this question using a two-sample t-interval.
(i) Using Minitab, find either the sample standard deviations of the
two variables…
5. (a) State the Residue Theorem. Your answer should include all the conditions required
for the theorem to hold.
(4 marks)
(b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the
anti-clockwise direction. Evaluate
に
dz.
You must check all of the conditions of any results that you use.
(5 marks)
(c) Evaluate
L
You must check all of the conditions of any results that you use.
ཙ
x sin(Tx)
x²+2x+5
da.
(11 marks)
3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula
for L(y).
(1 mark)
(b) Let UCC be open. Let f: U→C be continuous. Let y: [a,b] → U be a
contour. Suppose there exists a finite real number M such that |f(z)| < M for
all z in the image of y. Prove that
<
||, f(z)dz| ≤ ML(y).
(3 marks)
(c) State and prove Liouville's theorem. You may use Cauchy's integral formula without
proof.
(d) Let R0. Let w € C. Let
(10 marks)
U = { z Є C : | z − w| < R} .
Let f UC be a holomorphic function such that
0 < |ƒ(w)| < |f(z)|
for all z Є U. Show, using the local maximum modulus principle, that f is constant.
(6 marks)
Chapter 25 Solutions
Basic Technical Mathematics
Ch. 25.1 - Find an antiderivative of x3 + 4x.
Ch. 25.1 - Prob. 2PECh. 25.1 - Prob. 1ECh. 25.1 - Prob. 2ECh. 25.1 - Prob. 3ECh. 25.1 - Prob. 4ECh. 25.1 - In Exercises 5–12, determine the value of a that...Ch. 25.1 - Prob. 6ECh. 25.1 - Prob. 7ECh. 25.1 - Prob. 8E
Ch. 25.1 - Prob. 9ECh. 25.1 - In Exercises 5–12, determine the value of a that...Ch. 25.1 - Prob. 11ECh. 25.1 - Prob. 12ECh. 25.1 - Prob. 13ECh. 25.1 - Prob. 14ECh. 25.1 - Prob. 15ECh. 25.1 - Prob. 16ECh. 25.1 - Prob. 17ECh. 25.1 - Prob. 18ECh. 25.1 - Prob. 19ECh. 25.1 - In Exercises 13–40, find antiderivatives of the...Ch. 25.1 - Prob. 21ECh. 25.1 - Prob. 22ECh. 25.1 - Prob. 23ECh. 25.1 - Prob. 24ECh. 25.1 - Prob. 25ECh. 25.1 - Prob. 26ECh. 25.1 - Prob. 27ECh. 25.1 - Prob. 28ECh. 25.1 - Prob. 29ECh. 25.1 - Prob. 30ECh. 25.1 - Prob. 31ECh. 25.1 - Prob. 32ECh. 25.1 - Prob. 33ECh. 25.1 - Prob. 34ECh. 25.1 - Prob. 35ECh. 25.1 - Prob. 36ECh. 25.1 - Prob. 37ECh. 25.1 - Prob. 38ECh. 25.1 - Prob. 39ECh. 25.1 - Prob. 40ECh. 25.1 - Prob. 41ECh. 25.1 - Prob. 42ECh. 25.2 - Integrate: .
Ch. 25.2 - Prob. 1ECh. 25.2 - Prob. 2ECh. 25.2 - Prob. 3ECh. 25.2 - Prob. 4ECh. 25.2 - In Exercise 5–36, integrate each of the given...Ch. 25.2 - Prob. 6ECh. 25.2 - Prob. 7ECh. 25.2 - Prob. 8ECh. 25.2 - In Exercise 5–36, integrate each of the given...Ch. 25.2 - In Exercise 5–36, integrate each of the given...Ch. 25.2 - Prob. 11ECh. 25.2 - Prob. 12ECh. 25.2 - Prob. 13ECh. 25.2 - Prob. 14ECh. 25.2 - Prob. 15ECh. 25.2 - Prob. 16ECh. 25.2 - Prob. 17ECh. 25.2 - Prob. 18ECh. 25.2 - Prob. 19ECh. 25.2 - Prob. 20ECh. 25.2 - Prob. 21ECh. 25.2 - In Exercises 5–36, integrate each of the given...Ch. 25.2 - Prob. 23ECh. 25.2 - Prob. 24ECh. 25.2 - Prob. 25ECh. 25.2 - Prob. 26ECh. 25.2 - Prob. 27ECh. 25.2 - Prob. 28ECh. 25.2 - Prob. 29ECh. 25.2 - Prob. 30ECh. 25.2 - Prob. 31ECh. 25.2 - Prob. 32ECh. 25.2 - Prob. 33ECh. 25.2 - Prob. 34ECh. 25.2 - Prob. 35ECh. 25.2 - Prob. 36ECh. 25.2 - Prob. 37ECh. 25.2 - Prob. 38ECh. 25.2 - Prob. 39ECh. 25.2 - Prob. 40ECh. 25.2 - Prob. 41ECh. 25.2 - Prob. 42ECh. 25.2 - Prob. 43ECh. 25.2 - Prob. 44ECh. 25.2 - Prob. 45ECh. 25.2 - Prob. 46ECh. 25.2 - Prob. 47ECh. 25.2 - Prob. 48ECh. 25.2 - Prob. 49ECh. 25.2 - Prob. 50ECh. 25.2 - Prob. 51ECh. 25.2 - Prob. 52ECh. 25.2 - Prob. 53ECh. 25.2 - Prob. 54ECh. 25.2 - In Exercises 41–62, solve the given problems. In...Ch. 25.2 - Prob. 56ECh. 25.2 - Prob. 57ECh. 25.2 - Prob. 58ECh. 25.2 - Prob. 59ECh. 25.2 - Prob. 60ECh. 25.2 - Prob. 61ECh. 25.2 - Prob. 62ECh. 25.3 - Prob. 1PECh. 25.3 - Prob. 2PECh. 25.3 - Prob. 1ECh. 25.3 - Prob. 2ECh. 25.3 - Prob. 3ECh. 25.3 - Prob. 4ECh. 25.3 - Prob. 5ECh. 25.3 - Prob. 6ECh. 25.3 - Prob. 7ECh. 25.3 - Prob. 8ECh. 25.3 - Prob. 9ECh. 25.3 - Prob. 10ECh. 25.3 - Prob. 11ECh. 25.3 - Prob. 12ECh. 25.3 - Prob. 13ECh. 25.3 - Prob. 14ECh. 25.3 - Prob. 15ECh. 25.3 - Prob. 16ECh. 25.3 - Prob. 17ECh. 25.3 - Prob. 18ECh. 25.3 - Prob. 19ECh. 25.3 - Prob. 20ECh. 25.3 - Prob. 21ECh. 25.3 - Prob. 22ECh. 25.3 - Prob. 23ECh. 25.3 - In Exercises 15–24, find the exact area under the...Ch. 25.3 - Prob. 25ECh. 25.3 - Prob. 26ECh. 25.3 - Prob. 27ECh. 25.3 - Prob. 28ECh. 25.4 -
Evaluate: .
Ch. 25.4 - Prob. 2PECh. 25.4 - Prob. 1ECh. 25.4 - Prob. 2ECh. 25.4 - Prob. 3ECh. 25.4 - Prob. 4ECh. 25.4 - Prob. 5ECh. 25.4 - Prob. 6ECh. 25.4 - Prob. 7ECh. 25.4 - Prob. 8ECh. 25.4 - Prob. 9ECh. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - Prob. 12ECh. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - Prob. 15ECh. 25.4 - Prob. 16ECh. 25.4 - Prob. 17ECh. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - Prob. 20ECh. 25.4 - Prob. 21ECh. 25.4 - Prob. 22ECh. 25.4 - Prob. 23ECh. 25.4 - Prob. 24ECh. 25.4 - Prob. 25ECh. 25.4 - Prob. 26ECh. 25.4 - Prob. 27ECh. 25.4 - Prob. 28ECh. 25.4 - Prob. 29ECh. 25.4 - Prob. 30ECh. 25.4 - Prob. 31ECh. 25.4 - Prob. 32ECh. 25.4 - In Exercises 3–34, evaluate the given definite...Ch. 25.4 - Prob. 34ECh. 25.4 - In Exercises 35–54, solve the given problems.
35....Ch. 25.4 - Prob. 36ECh. 25.4 - In Exercises 35–54, solve the given problems.
37....Ch. 25.4 - Prob. 38ECh. 25.4 - Prob. 39ECh. 25.4 - Prob. 40ECh. 25.4 - Prob. 41ECh. 25.4 - Prob. 42ECh. 25.4 - Prob. 43ECh. 25.4 - Prob. 44ECh. 25.4 - Prob. 45ECh. 25.4 - Prob. 46ECh. 25.4 - Prob. 47ECh. 25.4 - Prob. 48ECh. 25.4 - Prob. 49ECh. 25.4 - Prob. 50ECh. 25.4 - Prob. 51ECh. 25.4 - Prob. 52ECh. 25.4 - In finding the average electron energy in a metal...Ch. 25.4 - Prob. 54ECh. 25.5 - Prob. 1PECh. 25.5 - Prob. 1ECh. 25.5 - Prob. 2ECh. 25.5 - Prob. 3ECh. 25.5 - Prob. 4ECh. 25.5 - Prob. 5ECh. 25.5 - Prob. 6ECh. 25.5 - Prob. 7ECh. 25.5 - Prob. 8ECh. 25.5 - Prob. 9ECh. 25.5 - Prob. 10ECh. 25.5 - Prob. 11ECh. 25.5 - Prob. 12ECh. 25.5 - Prob. 13ECh. 25.5 - Prob. 14ECh. 25.5 - Prob. 15ECh. 25.5 - Prob. 16ECh. 25.5 - Prob. 17ECh. 25.5 - Prob. 18ECh. 25.5 - Prob. 19ECh. 25.5 - Prob. 20ECh. 25.5 - Prob. 21ECh. 25.5 - Prob. 22ECh. 25.6 - Prob. 1PECh. 25.6 - Prob. 1ECh. 25.6 - Prob. 2ECh. 25.6 - Prob. 3ECh. 25.6 - Prob. 4ECh. 25.6 - Prob. 5ECh. 25.6 - Prob. 6ECh. 25.6 - Prob. 7ECh. 25.6 - Prob. 8ECh. 25.6 - Prob. 9ECh. 25.6 - Prob. 10ECh. 25.6 - Prob. 11ECh. 25.6 - Prob. 12ECh. 25.6 - Prob. 13ECh. 25.6 - Prob. 14ECh. 25.6 - Prob. 15ECh. 25.6 - Prob. 16ECh. 25.6 - Prob. 17ECh. 25.6 - Prob. 18ECh. 25 - Prob. 1RECh. 25 - Determine each of the following as being either...Ch. 25 - Prob. 3RECh. 25 - Prob. 4RECh. 25 - Prob. 5RECh. 25 - Prob. 6RECh. 25 - Prob. 7RECh. 25 - Prob. 8RECh. 25 - Prob. 9RECh. 25 - Prob. 10RECh. 25 - Prob. 11RECh. 25 - Prob. 12RECh. 25 - Prob. 13RECh. 25 - Prob. 14RECh. 25 - Prob. 15RECh. 25 - Prob. 16RECh. 25 - Prob. 17RECh. 25 - Prob. 18RECh. 25 - Prob. 19RECh. 25 - Prob. 20RECh. 25 - Prob. 21RECh. 25 - Prob. 22RECh. 25 - Prob. 23RECh. 25 - Prob. 24RECh. 25 - Prob. 25RECh. 25 - Prob. 26RECh. 25 - Prob. 27RECh. 25 - Prob. 28RECh. 25 - Prob. 29RECh. 25 - Prob. 30RECh. 25 - Prob. 31RECh. 25 - Prob. 32RECh. 25 - Prob. 33RECh. 25 - Prob. 34RECh. 25 - Prob. 35RECh. 25 - Prob. 36RECh. 25 - Prob. 37RECh. 25 - Prob. 38RECh. 25 - Prob. 39RECh. 25 - Prob. 40RECh. 25 - Prob. 41RECh. 25 - Prob. 42RECh. 25 - Prob. 43RECh. 25 - Prob. 44RECh. 25 - Prob. 45RECh. 25 - Prob. 46RECh. 25 - Prob. 47RECh. 25 - Prob. 48RECh. 25 - Prob. 49RECh. 25 - Prob. 50RECh. 25 - Prob. 51RECh. 25 - Prob. 52RECh. 25 - Prob. 53RECh. 25 - Prob. 54RECh. 25 - Prob. 55RECh. 25 - Prob. 56RECh. 25 - Prob. 57RECh. 25 - Prob. 58RECh. 25 - Prob. 59RECh. 25 - Prob. 60RECh. 25 - Prob. 61RECh. 25 - Prob. 62RECh. 25 - Prob. 63RECh. 25 - Prob. 64RECh. 25 - Prob. 65RECh. 25 - Prob. 66RECh. 25 - Prob. 67RECh. 25 - Prob. 68RECh. 25 - Prob. 1PTCh. 25 - Prob. 2PTCh. 25 - Prob. 3PTCh. 25 - Prob. 4PTCh. 25 - Prob. 5PTCh. 25 - Prob. 6PTCh. 25 - Prob. 7PT
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- 3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M a simple module? (b) State and prove Schur's Lemma for simple modules. (c) Let AM(K) and M = K" the natural A-module. (i) Show that M is a simple K-module. (ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a is a matrix in the centre of M, (K). [Recall that the centre, Z(M,(K)) == {a Mn(K) | ab M,,(K)}.] = ba for all bЄ (iii) Explain briefly why this means End₁(M) K, assuming that Z(M,,(K))~ K as K-algebras. Is this consistent with Schur's lemma?arrow_forward(a) State, without proof, Cauchy's theorem, Cauchy's integral formula and Cauchy's integral formula for derivatives. Your answer should include all the conditions required for the results to hold. (8 marks) (b) Let U{z EC: |z| -1}. Let 12 be the triangular contour with vertices at 0, 2-2 and 2+2i, parametrized in the anticlockwise direction. Calculate dz. You must check the conditions of any results you use. (d) Let U C. Calculate Liz-1ym dz, (z - 1) 10 (5 marks) where 2 is the same as the previous part. You must check the conditions of any results you use. (4 marks)arrow_forward(a) Suppose a function f: C→C has an isolated singularity at wЄ C. State what it means for this singularity to be a pole of order k. (2 marks) (b) Let f have a pole of order k at wЄ C. Prove that the residue of f at w is given by 1 res (f, w): = Z dk (k-1)! >wdzk−1 lim - [(z — w)* f(z)] . (5 marks) (c) Using the previous part, find the singularity of the function 9(z) = COS(πZ) e² (z - 1)²' classify it and calculate its residue. (5 marks) (d) Let g(x)=sin(211). Find the residue of g at z = 1. (3 marks) (e) Classify the singularity of cot(z) h(z) = Z at the origin. (5 marks)arrow_forward
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