Chapter 2: Complex Numbers
How to add, subtract, multiply and divide complex numbers in rectangular form
(Shaum's pg. 10)
Conversion between rectangular and polar coordinates
(pg. 48)
modulus of a complex number
(pg. 49)
real part, imaginary part, modulus, and angle of a complex number
(pg. 49)
Complex Conjugate of a complex number
(pg. 50)
complex conjugate in polar form
(pg. 50)
How to find the modulus or absolute value of a complex number, using the complex conjugate
(pg. 53)
Do problem 2.6.1
(pg. 57)
What's i squared? To the power of 4? To a power congruent to 2 mod 4? Congruent to 0 mod 4?
-1 and 1
What is an inductor?
A circuit element with a high inductance. It will oppose changes in current. If we connect a battery in series with an inductor, the current will increase gradually rather than instantaneously. Likewise, if we decrease the current in a circuit with a inductor, the current will decrease gradually rather than instantaneously (review)
What is the disk of convergence of a complex power series?
The disk in the complex plane in which the series is convergent (pg. 58)
What is the complex conjugate of the quotient of two complex numbers?
The quotient of their complex conjugates. This is readily proven if you write the numbers in polar form (problem 2.5.25)
What is the radius of convergence of a complex power series?
The radius of the disk of convergence (pg. 58)
DeMoivre's Theorem
This can be used to find the double, triple, quadruple, ... angle formulas for sine and cosine (pg. 64)
what is e^(iπ/2)
Using Euler's formula we get i
Properties of complex exponentials
(pg. 60)
Euler's Formula
(pg. 61)
How to take the nth root of a complex number
(pg. 64)
Steps to finding the n nth roots of a complex number
(pg. 66)
Inverse trig and hyperbolic functions
(pg. 74)
What is inductance?
A measure of the opposition to a change in current. It has units of Henries 1H = 1V.s/A (review)
sine and cosine in terms of complex exponents
FIXME: how are these derived? (pg. 68)
T/F the imaginary part of a complex number is imaginary
False, the imaginary part of the complex number a + bi is b which is a real scalar (pg. 47)
Explain how to multiply and divide complex numbers (hint use the polar form)
For multiplication, multiply their lengths and add their polar angles. For division, divide their lengths and subtract their polar angles (pg. 62)
What are complex infinite series?
Infinite series where the terms are complex (pg. 56)
Define capacitive reactance
It represents opposition to the flow of charge. We might think of it as the resistance of a capacitor in an AC circuit. It is inversely proportional to the angular frequency of the AC circuit and inversely proportional to the capacitance
What is inductive reactance?
It represents opposition to the flow of charge. We might think of it as the resistance of an inductor. It is directly proportional to the angular frequency of the AC circuit and directly proportional to the inductance
How to write e^z (z complex) in terms of sines and cosines
Just use the multiplication property of exponents and Euler's formula (pg. 67)
What are complex numbers?
Numbers of the form a + bi where a and b are real scalars. Thus 3i, 2 + 3i, and 2 are all complex numbers. The first scalar, a, is called the real part and the second scalar, b, is called the imaginary part (pg. 46)
What are complex power series?
Power series in which the coefficients and variables are complex (pg. 58)
How to take logarithms of complex numbers
Put the complex number into its polar form and use the properties of logarithms to split it into a real part and an imaginary part (pg. 72)
How to take powers of complex numbers
This is readily derived if we use the rules for multiplying and dividing complex numbers. Be sure to use polar form (pg. 64)
Series definition of the exponential function e^z when our domain is the complex plane.
This series converges for all z. (pg. 60)
Relate the voltage and current of a resistor
Voltage is proportional to current (pg. 77)
How do we test to see if a complex infinite series converges?
We test to see that both the real and imaginary parts of the series converge (pg. 56)
rectangular form vs. polar form of a complex number
We typically do additional and subtraction in rectangular form. While we typically do multiplication and division in polar form pg. 48
When to use radians vs. degrees
You can use degrees to and and subtract angles as long as the final step is to find the sine, cosine, or tangent of the resulting angle. However, in formulas, we must use radians (pg. 49)
How do you find the complex conjugate of a complex expression?
You change the sign in front of all the i terms and take the complex conjugate of any complex variables (pg. 53)
What's the geometric effect of squaring a complex number?
You square the length and double the polar angle. This is easy to see if the complex number is in polar form
Relate the voltage and current of a capacitor
current is proportional to the time derivative of the voltage. Alternatively, the voltage is proportional to the time integral of the current (pg. 77)
Power series for sine and cosine
fixme: how are these derived??? (pg. 61)
Definitions of sin(z) and cos(z) with z complex
pg. 68
Prove that sin²(z) + cos²(z) = 1
pg. 69
Show that the derivative of sin(z) = cos(z)
pg. 69
Definitions of the hyperbolic functions
pg. 70
Properties of complex conjugates
practice proving these
Some hyperbolic identities. Relate sinhz to sinz and coshz to cosz
sinhz = =isin(iz) coshz = cos(iz)
If the position of a particle is given by some complex function z(t) = x+iy, how do we find the velocity and acceleration?
v = dz/dt and a = d²z/dt² (see example in section 2.16 pg. 76)
Relate the voltage and current of an inductor
voltage is proportional to the time derivative of the current (pg. 77)
The resonance frequency for both series and parallel RLC circuit
ω₀=1/√(LC) (problems 2.16.8, 2.16.9 pg. 78)
