www.whyslopes.com
Volume 3, Why Slopes and More Math
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YOU are better than YOU think. Show yourself how:
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Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
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Caution: Site advice is approximately
correct, for some circumstances, not all. . That leaves room for thought and
refinement.. |
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Online Volume 2, Three Skills
for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key skills
and concepts, those needed in calculus, again to make the hard easier. A visual
understanding of complex numbers
may serve as back ground info for partial fraction decomposition.
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Chapter 24
Complex Logs, Powers and Exponentials
This last chapter defines (states formulas for) the
exponential, logarithms and powers of complex numbers
x+iy etc. If you are a science and engineering student
you will eventually meet these functions and see their
properties. This chapter gives a list of functions which
you should expect to meet and understand in the first two
years of your university studies. (The further discussion
of these functions is left to a second or third course on
calculus. From time to time, you should refer to the
definitions given below to see how many on this list remain
to be seen in your courses.)
The exponential of a complex number x+iy
is given by
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exp(x+iy) = ex[cos(y)+isin(y)] = ex
cis(y) |
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Further, if x+iy = r cos(q) +isin(q) ¹ 0
with - = -180° < q £ 180° = p then the
principal value of the natural logarithm
This definition implies
exp(2pi + ln(x+iy)) = x+iy for each integer n.
Note also:
- fundamental properties of exponentials:
exp(z1+z2) = exp(z1)exp(z2)
- fundamental property of logarithms: ln(z1z2) = ln(z1)+ln(z2)+i2pn for some integer n Î {0,±1},
- first inverse property: exp(ln(z)) = z if z ¹ 0,
- second inverse property: ln(exp(z))-z = 2npi
for some integer n,
- powers defined: zx+iy = exp((x+iy)ln(z)) for z ¹ 0,
- the definition of a logarithm to the complex base a+ib:
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loga+ib(z) = |
ln(z)
ln(a+ib) |
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- the hyperbolic cosine of the complex number x+iy defined:
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cosh(x+iy) = |
exp(x+iy)+exp(-x-iy)
2 |
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What do you get if y = 0? What do you get if x = 0?
- The hyperbolic sine of the complex number x+iy defined:
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sinh(x+iy) = |
exp(x+iy)-exp(-x-iy)
2i |
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What do you get if y = 0. What do you get if x = 0?
Note that for real number A, we can easily show that
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cos(A) = |
exp(iA)+exp(-iA)
2 |
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and that
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sin(A) = |
exp(iA)-exp(-iA)
2i |
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follow from the definition of the exponential function
The above two identities are consistent with more generally
letting
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cos(A+iB) = |
exp(i(A+iB))+exp(-i(A+iB))
2 |
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and
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sin(A+iB) = |
exp(i(A+iB))-exp(-i(A+iB))
2i |
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for each complex number a+iB as well: what happens when B = 0?
Two Problems: How are the definitions of the
cosine and hyperbolic cosine related? How are the definitions of
the sine and hyperbolic sine related? | |
For help in calculus, explore
Volumes
2. Three Skills
for Algebra
and 3. Why
Slopes & More Math, and Calculus
Introduction site area. See how to learn or teach key skills and
concepts, some not all.
Foreword, One Calculus preview and Online Chapters:
(V) signals video (RealPlayer Format) to
watch
Area Intro
Foreword
Chapter Descriptions
1. Introduction
2. Calculus Starter Lesson
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review (optional)
9 On Calculus Studies
11 Slope of Slope
13 Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17 What is Area
18 Integration
18 Area Calculation
18 Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity Motions
10 Slopes without Units.
10 Units & Slopes
10 Units in Cost vs. Quantity
10 How Units Appear
10 Unit Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Content Guide
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
Theorem
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
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