Chapter 24. Complex Logarithms, Powers and Exponentials
Volume 3, Why Slopes and More Math.
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. $\def\cis{\mbox{cis}}$
The exponential of a complex number $x+iy $ is given by
\[\exp(x+iy)=e^x[\cos(y)+i\sin(y)]=e^x\cis(y) \]
Further, if
\[x+iy =r \cos(\theta) +i\sin(\theta) \ne 0 \]
with
\[-\pi =-180^\circ < \theta \le 180^\circ=\pi\]
then the principal value of the natural logarithm
\[\ln(x+iy)=\ln(r)+i\theta. \]
This definition implies \[\exp(2\pi i + \ln(x+iy)) =x+iy\]
for each integer $n.$
Note also:
-
fundamental properties of exponentials:
\[\exp(z_1+z_2)=exp(z_1)\exp(z_2)\]
-
fundamental property of logarithms:
\[\ln(z_1z_2) = \ln(z_1)+\ln(z_2) +i2\pi n\]
for some integer $n \in \{0,\pm 1\}$,
-
first inverse property:
\[\exp(\ln (z))=z\]
if $z \ne 0$,
-
second inverse property:
\[\ln(\exp(z))-z = 2n\pi i\]
for some integer $n$,
-
powers defined:
\[z^{x+iy}=\exp((x+iy)\ln(z))\]
for $z \ne 0$
-
the definition of a logarithm to the complex base $a+ib$
\[\log_{a+ib}(z)=\frac{\ln(z)}{\ln(a+ib)}\]
-
the hyperbolic cosine of the complex number $x+iy$ defined:
\[\cosh(x+iy)=\frac{\exp(x+iy)+exp(-x-iy)}2\]
What do you get if $y=0$? What do you get if $x=0$?
-
The hyperbolic sine of the complex number $x+iy$ defined:
\[\sinh(x+iy)=\frac{\exp(x+iy)-exp(-x-iy)}{2i}\]
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
\[\cos(A)=\frac{\exp(iA)+\exp(-iA)}{2}\]
and that
\[\sin(A)=\frac{\exp(iA)-\exp(-iA)}{2i}\cdot\]
follow from the definition of the exponential function The above two
identities are consistent with more generally letting
\[\cos(A+iB)=\frac{\exp(i(A+iB))+\exp(-i(A+iB))}{2}\]
and
\[\sin(A+iB)=\frac{\exp(i(A+iB))-\exp(-i(A+iB))}{2i}\]
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?
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Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
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protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
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Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
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work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
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Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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