Integrating Factors
When the equation
Pdx + Qdy = 0 (1)
is not exact, we may seek to make it exact by multiplying throughout by a suitable notation μ(x,y). Applying the condition for the equation
(μP)dx + (μQ)dy = 0
to be exact, we see that such a function μ must satisfy the equation
(2)
If we can find a function μ which satisfies this equation, we may say that we have found an integrating factor of the equation (1).
It is not an easy matter to find functions which satisfy equation (2) unless we make some simplifying assumption about the nature of μ. μ is a function of x alone. If we assume that μ = μ(x) is a function of x alone,
the equation (2) reduces to
which may be written in the form
(3)
Now the
function on the left-hand side of this equation contains x
alone and this must therefore also be true of the function on the
right-hand side of this equation. Our assumption that μ
is a function of x
alone is justified, therefore, only if it turns out that 
is a
function of x
alone. If we denote this function by ψ(x),
equation (3) may be put in the form 
showing that log μ(x)= ∫ψ(x)dx (4)
is an equation from which we can easily determine μ(x).
*Source: J.G.Deferes and Sheddon The Mathematics of Medicine and Biology, p.446-447.
Unit 3.
COMPLEX NUMBERS
Pre-reading tasks.
General comprehension questions:
What kind of number is called complex?
Can it be represented geometrically? In what form? Why is it useful?
What operations can be done with complex numbers?
A list of words you might need for the answer:
Field, subfield, imaginary unit, argument, complex plane, conjugate, addition, subtraction, multiplication, division, to obey the rule/law
Learn to read the following formulas:
1) x2=c
x squared is equal to c;
2)
the square root of c;
3) -
minus square root of c;
4) x2+2bx+c=0
x squared plus two bx plus c is equal to 0;
5) (a+b)2=b2-c
a plus b in parenthes, all to the second power, is equal to b squared minus c.
or: parenthesis, a plus b, close parenthesis, to the second power is equal to b squared minus c;
6) b≥a
b is greater than or equal to a;
7) b<a
b is less than a;
8) (-a)(-b)=ab
minus a multiplied by minus b is equal to ab;
9) b≠0
b is not equal to 0;
10) αβ=(α+bi)(c+di)=ac-bd+(ad+bc)i
α times β is equal to α+bi in parentheses, all multiplied by c+di, in parentheses, is equal to ac minus bd plus parenthesis ad plus bc close parentheses, multiplied by i;
11) γ=α/β
γ is equal to α divided by β; or: γ is equal to α over β;
12)
|α|²+
+
+|β|2
modulus of α squared plus α β barred plus α barred β plus modulus of β squared;
13) |α1…αn|
modulus of α sub 1 and so on α sub n;
14) |αn|=|α|n
modulus of α to the power n is equal to modulus of α all to the power n.
Read Text 1 aloud with the formulas:
Text 1.
The complex
numbers are the field
of numbers of
the form
,
where
and
are
real
numbers
and i
is the imaginary
unit
equal to the square
root
of
,
.
When a single letter
is
used to denote a complex number, it is sometimes called an "affix."
In component notation,
can
be written
.
The field
of complex numbers includes the field
of real
numbers
as a subfield.
Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. However, recognition of this fact is one that took a long time for mathematicians to accept. For example, John Wallis wrote, "These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen) are reputed to imply that the Case proposed is Impossible" (Wells 1986, p. 22).
Through the Euler formula, a complex number
|
(1) |
may be written in "phasor" form
|
(2) |
Here,
is known as the complex
modulus
(or sometimes the complex norm) and
is known as the complex
argument
or phase.
The plot above shows what is known as an Argand
diagram
of the point
,
where the dashed circle represents the complex
modulus
of
and the angle
represents
its complex
argument.
Historically, the geometric representation of a complex number as
simply a point in the plane was important because it made the whole
idea of a complex number more acceptable. In particular, "imaginary"
numbers became accepted partly through their visualization.
Unlike real numbers, complex numbers do not have a natural ordering, so there is no analog of complex-valued inequalities. This property is not so surprising however when they are viewed as being elements in the complex plane, since points in a plane also lack a natural ordering.
The
absolute
square
of
is
defined by
,
with
the complex
conjugate, and the argument may be
computed from
|
(3) |
The real
and
imaginary
parts
are
given by
|
|
|
(4) |
|
|
|
(5) |
|
|
|
(6) |
|
|
|
(7) |
de
Moivre's identity
relates powers
of complex numbers for real
by
|
(8) |
A power of complex number to a positive integer exponent can be written in closed form as
|
(9) |
The first few are explicitly
|
|
|
(10) |
|
|
|
(11) |
|
|
|
(12) |
|
|
|
(13) |
(Abramowitz and Stegun 1972).
Complex addition
|
(14) |
complex subtraction
|
(15) |
complex multiplication
|
(16) |
and complex division
|
(17) |
can also be defined for complex numbers. Complex numbers may also be taken to complex powers. For example, complex exponentiation obeys
|
(18) |
where
is
the complex
argument.
*Source: Weisstein, Eric W. "Complex Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ComplexNumber.html