16.3 The Fundamental Theorem for Line Integrals
1. Quote. There again, that is a fundamental principle: no two situations are alike.
(Lakhdar Brahimi, Algerian Diplomat, 1934 - )
2. Reminder. The Fundamental Theorem of Calculus: If F is continuous on [a,
14.1 Line Integral in the Complex Plane
1. Quote. Knowledge is of no value unless you put it into practice.
(Anton Pavlovich Chekhov, Russian physician, dramaturge, and author, 1860-1904)
2. Goal. Show that
x2
e
0
b2
cos 2bxdx =
e .
2
3. The Same But A L
16.1 Vector Fields
1. Quote. To a mathematician the term vector eld denotes a slightly fancier construction. A vector
eld X on a manifold M is a smooth section of the tangent bundle T M , that is for each point m M
a choice of a tangent vector X(x) Tm M s
16.7 Surfaces Integrals
1. Quote. The good life is one inspired by love and guided by knowledge.
(Bertrand Arthur William Russell, 3rd Earl Russell, British philosopher, logician, mathematician, historian, and social critic, 1872 - 1970)
2. Problem. Find
14.4 Derivatives of Analytic Functions
1. Quote. If Im free, its because Im always running.
(Jimi Hendrix, American musician, singer and songwriter, 1942 -1970)
2. Goal. Show that
b2
cos 2bxdx =
e .
2
x2
e
0
3. Derivatives of an Analytic Function. If f (
14.2 Cauchys Integral Theorem
1. Quote. Change your opinions, keep to your principles; change your leaves, keep intact
your roots.
(Victor Marie Hugo, French poet, novelist, and dramatist, 1802 -1885)
2. Goal. Show that
x2
e
0
b2
cos 2bxdx =
e .
2
3. Rem
15.2 Power Series
1. Quote. Time is not a line, but a series of now-points.
(Taisen Deshimaru, Japanese Buddhist teacher, 1914 - 1982)
2. Problem. For which z C does the power series
1 + z + z2 + z3 + . . .
converge?
3. Power Series. A power series in pow
15.1 Sequence, Series, Convergence Tests
1. Quote. Never forget that the most powerful force on earth is love.
(Nelson Aldrich Rockefeller, American businessman, philanthropist, public servant,
and politician, 1908 - 1979)
2. Problem. What is
i
1+ +
2
i
2
14.3 Cauchys Integral Formula
1. Quote. Of course there is no formula for success except perhaps an unconditional
acceptance of life and what it brings.
(Arthur Rubinstein, Polish-American classical pianist, 1887 - 1982)
2. Goal. Show that
x2
e
0
b2
cos
15.4 Taylor and Maclaurin Series
1. Quote. I intend to live forever. So far, so good.
(Steven Wright, American comedian, actor and writer, 1955-)
2. Problem. Represent the given analytic function f (z) by a power series.
3. Taylor Series. The Taylor serie
13.7 Logarithm. General Power. Principal Value
1. Quote. The less eort, the faster and more powerful you will be.
(Bruce Lee, Chinese American martial artist, actor, philosopher, and lmmaker, 19401973)
2. Problem. Solve the equation ez = i.
3. Denition. F
13.6 Trigonometric and Hyperbolic Functions. Eulers Formula
1. Quote. The imaginary is what tends to become real.
(Andr Breton, French poet, 1896-1966)
e
2. Problem. What is sin i?
3. Denition. For z C we dene
1
cos z = (eiz + eiz )
2
sin z =
1 iz
(e eiz
16.6 Parametric Surfaces and Areas
1. Quote. When a blind beetle crawls over the surface of a curved
branch, it doesnt notice that the track it has covered is indeed curved.
I was lucky enough to notice what the beetle didnt notice.
(Albert Einstein, Germ
16.4 Greens Theorem
1. Quote. In the Press, and shortly will be published, by subscription, An
Essay on the Application of Mathematical Analysis to the Theories of
Electricity and Magnetism. By George Green. Dedicated (by permission), to his Grace the Duk
16.8 Stokes Theorem
1. Quote. I too feel that I have been thinking too much of late, but in
a dierent way, my head running on divergent series, the discontinuity
of arbitrary constants, . I often thought that you would do me good
by keeping me from being
13.1 Complex Numbers
1. Quote. Nothing is perfect. Life is messy. Relationships are complex. Outcomes are
uncertain. People are irrational.
(Hugh Mackay, Australian Social Researcher, 1945 )
2. Problem. Solve x2 = 1 for x.
3. Complex Numbers. The eld of c
16.9 The Divergence Theorem
1. Quote. There is an inevitable divergence between the world as it is
and the world as men perceive it.
(James William Fulbright, 1905-1995, was a United States Senator)
2. Problem. Find the ux of the vector eld F(x, y, z) =
13.3 Derivative. Analytic Function
1. Quote. Nothing is built on stone; all is built on sand, but we must build as if the
sand were stone.
(Jorge Francisco Isidoro Luis Borges, Argentine writer, 1899 - 1986)
2. Problem. Is the function f (z) = z dierentia
13.5 Exponential Functions.
1. Fact. In a 1988 poll, readers of the journal Mathematical Intelligencer chose Eulers
equation
ei + 1 = 0
as the single most beautiful equation in all of mathematics.
2. Problem. If z C, what does ez mean?
3. Denition. For z
13.4 Cauchy-Riemann Equations. Laplaces Equation.
1. Quote. What we know is not much. What we do not know is immense.
(Pierre-Simon Laplace, French mathematician and astronomer, 1749-1827)
2. Problem. Is the function f (z) = ln |z| + iArg z analytic?
3. R
13.2 Polar Form of Complex Numbers
1. Quote. The works must be conceived with re in the soul but executed with clinical
coolness.
(Joan Mir i Ferr`, Spanish painter, 1893 - 1983)
o
a
2. Problem. What is 3 i?
3. Modulus and Argument. Let z = x + iy C\cfw_(
16.3 Residue Integration Method
1. Quote. Love is so short, forgetting is so long.
(Pablo Neruda, Chilean poet, diplomat and politician, 1904-1973)
2. Two Reminders.
(a) If z0 is an isolated singular point of f (z) then f (z) has a Laurent series
an (z z0
16.2 Singularities and Zeros. Innity
1. Quote. An innity of passion can be contained in one minute, like a crowd in a small
space.
(Gustave Flaubert, French writer, 1821 - 1880)
2. Problem. What can we say about
lim f (z)
z0
if
(a) f (z) =
1
z
(b) f (z) =
Assignment 6
Solutions
Section 13.5
4. Find Re z, Im z, and |z| if z = e0.61.8i .
Solution: By denition
z = e0.61.8i = e0.6 (cos(1.8) + i sin(1.8) = e0.6 (cos 1.8 i sin 1.8).
Hence
Re z = e0.6 cos 1.8, Im z = e0.6 sin 1.8, and |z| = e0.6 .
10. Write i and
Assignment 7
Solutions
Section 13.3
8. Determine and sketch the set of all complex number z such that
|z + i| |z i|.
Solution. Let z = x + iy. Then z + i = x + i(y + 1), z i = x + i(y 1)
and
|z + i| |z i|
|z + i|2 |z i|2
x2 + (y + 1)2 x2 + (y 1)2
y 2 + 2
Assignment 8
Section 14.3
12. Integrate counterclockwise. Show the details.
C
z2
z
dz,
+ 4z + 3
C is the circle with centre -1 and radius 2.
Solution: We note that
C
z2
z
dz =
+ 4z + 3
C
z
dz
(z + 1)(z + 3)
and that
| 3 (1)| = 2.
z
is not dened at the poi
Assignment 9
Solutions
Section 15.1
2. Is the sequence zn = (3 + 4i)n /n! bounded? Convergent? Find its
limit points. Show your work in detail.
Solution: We note that |3 + 4i| = 5. Let = arctan(4/5). Then
3 + 4i = 5(cos + i sin ) and
zn =
(5(cos + i sin )
Assignment 10
Solutions
Section 15. 3
Find the radius of convergence in two ways: (a) directly by the CauchyHadamard formula in Sec 15.2, and (b) from a series of simpler terms by
using Th 3 or Th 4.
5n
zn
8.
n(n + 1)
n=1
5n
Solution: (a) From an =
by the
1/2
ENSC120 Lab Exercise 5
Measurement of Phase in sinusoidal signals
This final lab exercise will instruct you how to measure the phase difference between two signals observed using
the oscilloscope. Follow the steps as described below.
R = 10K
Input
Out
Lab2 Experiment and Record submission
Do not start this experiment before you have gone through the Scope Video Tutorial and the
Function Generator Operation instructions
1. Open the Lab2_Record.pdf using FOXIT.
2. Set the FnGen to default settings [See s
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