Transcript Tricks with Complex Number

Tricks with Complex Number
phys3330 – Spring 2012
Things you need to know about the complex number for this course:
1.Perform algebraic operations on complex number and represent a given
complex number graphically and express it in polar form.
2. Represent a sinusoidal function as the real and imaginary part of an
exponential and use this representation for adding trigonometric functions.
3. Set up a linear differential equation to describe the behavior of LCR circuit
and subject to a sinusoidal applied voltage.
4. Use complex exponentials to solve homogenous and inhomogeneous linear
differential equation with constant coefficient.
1. Complex number - The imaginary unit:
and
(where x and y are REAL number)
(a)General complex number
real part of z, x= Re z
imaginary part of z, x= Im z
(b) Arithmetic in complex number : just like ordinary
+
−
×
÷
(c) Complex conjugate (z → z*) : Replacing i to -i
✱
→
✱
✱
✱
(used in rationalize the denominator)
→ the modulus of z
2. Power Series for exponential and trigonometric functions:
Now compare trigonometric and hyperbolic function in complex number:
3. Polar representation of a complex number z=x+iy
y (imaginary)
Representing z=x+iy by the point (x,y)
z=x+iy
Then,
r
ϕ
where,
x (real)
Now we can always write
x
✵Polar representation is advantageous for multiplication and division!
Let
then
✵Example. Find z satisfying z3 = -8i
(solution) Since
they all have modulus 2. Thus
Now,
thus,
So,
and,
y
Homework Problems
1. Let
(a) Represent z1 and z2 in the complex plane and find their real and imaginary part
(a) Evaluate z1 + z2 and
2. By writing out cos θ in terms of exponentials and using the binomial expansion,
express (cos θ)5 in terms of cos θ, cos 3θ and cos 5θ.
3. Evaluate the sum
4. Suppose that frequencies
express the sum
and
differ only slightly. Use the complex exponential,
(A is a constant) in the form of
where A(t) is a slowly varying function of time.