Tuesday 7 June 2016
Complex numbers
A complex number is a number that can be expressed in the form a + bi, where a and bare real numbers and i is the imaginary unit, that satisfies the equation i2 = −1.In this expression, a is the real part and b is the imaginarypart of the complex number.
Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.
As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics,electrical engineering, and statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century.
Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation
has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the imaginary unit i where i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are−3 + 3i and −3 − 3i, as can be verified using the fact that i2 = −1:
According to the fundamental theorem of algebra, all polynomial equational with real or complex coefficients in a single variable have a solution in complex numbers.
Definition:-
A complex number is a number of the form a+ bi, where a and b are real numbers and i is the imaginary unit, satisfying i2 = −1. For example, −3.5 + 2i is a complex number.
The real number a is called the real part of the complex number a + bi; the real numberb is called the imaginary part of a + bi. By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginary part.The real part of a complex number z is denoted by Re(z) orℜ(z); the imaginary part of a complex number z is denoted by Im(z) or ℑ(z). For example,
Hence, in terms of its real and imaginary parts, a complex number z is equal to . This expression is sometimes known as the Cartesian form of z.
A real number a can be regarded as a complex number a + 0i whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi whose real part is zero. It is common to write a for a + 0i and bifor 0 + bi. Moreover, when the imaginary part is negative, it is common to write a − bi withb > 0 instead of a + (−b)i, for example 3 − 4iinstead of 3 + (−4)i.
The set of all complex numbers is denoted by ℂ,
Addition:-
To add two complex numbers we add each part separately:
(a+bi) + (c+di) = (a+c) + (b+d)i
Multiplying
To multiply complex numbers:
Each part of the first complex number gets multiplied by
each part of the second complex number
each part of the second complex number
Just use "FOIL", which stands for "Firsts, Outers,Inners, Lasts"
| |
(a+bi)(c+di) = ac + adi + bci + bdi2
| |
Like this:
But There is a Quicker Way!
Use this rule:
(a+bi)(c+di) = (ac−bd) + (ad+bc)i
1. Add: (7 + 5i) + (8 - 3i)
2. Add: (2 + 3i) + (-8 - 6i)
3. Express the sum of 
and 
in the form 
.
4. Add 
and 
.
Example: i2
i can also be written with a real and imaginary part as 0 + i
Complex Plane
We can also put complex numbers on a Complex Plane.- The Real part goes left-right
- The Imaginary part goes up-down
| i2 = (0 + i)2 | = (0 + i)(0 + i) | ||
| = (0×0 − 1×1) + (0×1 + 1×0)i | |||
| = −1 + 0i | |||
| = −1 |
Prove that
for any integer n.
Corollary 1.2.15: DeMoivre's Formula For any integer n and any real number twe have
(cos(t) + i sin(t))n = cos(nt) + i sin(nt)Proof
DeMoivre's Formula is quite something. It says that if you take a number on the unit circle (i.e. with lenght 1) with initial argument (angle) t and multiply it by itself, it simply rotates around the unit circle by that angle t. Each time you multiply the number by itself, the vector rotates another tdegrees. In other words, in this case the power operator results in a simple rotation.
Powers of a vector z with |z|=1
Two interesting questions related to this rotation, taken from the field of Complex Dynamics, are: suppose z is a complex number with |z|=1. Then:
- find conditions for Arg(z) such that zn = z for some integer n. Such a point, incidentally, is called periodic of order n.
- if Arg(z)/ is irrational, what can you say about the sequence {z, z2, z3, z4, ...}? Does it, for example, converge? Such a sequence, incidentally, is called the orbit of z.
Proposition 1.2.16: Finding Roots For any positive integer n and any non-zero complex number a = r cis(t) the equation zn = a has exactly n distinct roots given by:
z =where k = 0, 1, 2, ... n-1.
In a previous example we found the two square roots of i, which turned out to be a fair amount of work. The above proposition allows us to dispense with such a question quickly. For example, the two solutions for
z2 = i = cis(/2)are:
z1 = cis(/2/2) = cis(/4)
z2 = cis((/2 + 2)/2) = cis(5/4)
which you can quickly check using DeMoivre's Formula. Here is a geometric interpretation of this proposition: let's find, for example, the three third-roots of i, i.e. we want to find all solutions toz3 = i.
1: Draw the vector i 2: Divide angle by 3 for first root 3: Draw 3 equally spaced segments,
starting at the first root
Note, in particular, that the third root of i turned out to be -i, which indeed checks out:
(-i)(-i)(-i) = i*i*(-i) = (-1)*(-i) = i
This proposition is very satisfying: it says that at least every simple polynomial equation of degreen has n solutions. Later we will see that this is true in general: every n-th degree polynomial hasn roots, no "if's" and "but's". This, in fact, is a sign of things to come: many theorems in complex analysis will turn out to be very "satisfying" and nicely structured, which is one reason that the study of complex analysis is a lot of fun (I think -:). But first a few more 'profane' examples.
Example 1.2.17: Finding roots geometrically
Find the cube roots of 8 and iand draw them.
Let's conclude this chapter with a result that illustrates what a 'nicely structured' theorem in complex analysis can look like.
Proposition 1.2.18: Roots of Unity The n n-th roots of unity are given bywnk, where k = 0, 1, 2, ... n-1 and
wn = cis(2/n)They form the vertices of a regular polygon and add up to zero, i.e. they satisfy the equation:
1 + wn + wn2 + ... + wnn-1 = 0
This is neat: not only does the equation zn = 1have exactly n solutions (one of which is, of course, z=1), but the solutions have this really pretty geometric structure of forming a regular polygon, which implies algebraically that they add up to zero as vectors. Here are, for example, the eight roots of z8=1:
You can see their regular structure. When you add them all as vectors, you indeed get the zero vector.
(.Ref from Wikipedia and Mathisfun private.Shu.edu)
Friday 27 May 2016
Tuesday 17 May 2016
differentibily
We present a new proof of the differentiability of exponential functions. It is based
entirely on methods of differential calculus. No current or recent calculus text gives or
cites a proof of the differentiability that depends only on such elementary tools. Our
proof makes it possible to give a comprehensive treatment of the derivative properties
of exponential and logarithmic functions in that order in differential calculus, building
on the standard introduction to these topics in precalculus courses. This is the logical
order and has considerable pedagogical merit.
Most calculus books defer the treatment of exponential and logarithmic functions to
integral calculus in order to prove differentiability. A few texts introduce these topics in
differential calculus under the heading of “early transcendentals” but defer the proof of
differentiability to integral calculus. Both approaches have serious pedagogical faults,
which are discussed later in this paper.
Our proof that exponential functions are differentiable provides the missing link
that legitimizes the “early transcendentals” presentation.
Preliminaries
We assume that ar has been defined for a > 0 and r rational in a precalculus course
and that the familiar rules of exponents are known to hold for rational exponents. It is
natural to define ax for a > 0 and x irrational as the limit of ar as r → x through the
rationals. In this way, ax is defined for all real x.
Basic properties of ax for real x are inherited by limit passages from corresponding
properties of ar for r rational. These properties include the rules of exponents with
real exponents and
ax is positive and continuous,
ax is increasing if a > 1,
ax is decreasing if a < 1.
It is not especially difficult to justify the definition of ax for x irrational and to
derive the foregoing properties of ax for x real, but there are a lot of small steps. A
program along these lines is carried out by Courant in [2, pp. 69–70]. The general idea
of each step is well within the grasp of students in typical calculus classes. However,
just as properties of ar with r rational are routinely stated without proof, it is better to
give just an overview of the basic properties of ax with x real, illustrated with graphs,
and move on to the question of differentiability, which is more central to differential
calculus.
A more complete development, beginning with the derivation of properties of ar
with r rational, might be given in an honors class. The properties can be extended to
ax with x real with the aid of the density of the rationals in the reals and the squeeze
laws for limits. The conclusion that ax with a > 1 is increasing also relies on the
following proposition which should seem evident from graphical considerations:
If f is a continuous function on a real interval I
and f is increasing on the rational numbers in I,
then f is increasing on I.
The same proposition will provide a key step in the proof that ax is differentiable.
Henceforth, we restrict our attention to properties of ax with a > 1. Corresponding
properties of ax with 0 < a < 1 follow from ax = (1/a)
−x .
The differentiability of ax
Consider an exponential function ax with any a > 1. In order to prove that ax is differentiable
for all x, the main task is to prove that it is differentiable at x = 0. Our proof
of this depends only on methods of differential calculus. It is motivated by the fact
that the graph of ax (see Figure 1) is concave up, even though this fact is not assumed
a priori
 |
| figure 1 |
Graph of ax with B = (x, ax ) and C = −x, a−x for x > 0
In Figure 1, imagine that x → 0 with x > 0 and x decreasing. Then B and C slide
along the curve toward A. The upward bending of the curve seems to imply that
slope AB decreases, slope AC increases,
and slope AB − slope AC → 0.
It follows that the slopes of AB and AC approach a common limit, which is the slope of
the tangent line T in Figure 1 and the derivative of f (x) = ax at x = 0. This geometric
argument will be made rigorous.
The curve in Figure 1 is actually the graph of f (x) = 2x . The following table gives
values of the slopes of AB and AC rounded off to two decimal places. It appears that
the slopes of AB and AC approach a common limit, which is f
(0) = slope T ≈ 0.7.
x 1 1/2 1/4 1/8 1/16 1/32
slope AB 1 .83 .76 .72 . 71 .70
slope AC .50 .59 .64 . 66 .67 .69
With this preparation, we are ready to prove that f (x) = ax is differentiable at
x = 0. The foregoing geometric description of the proof and the numerical evidence
should be informative and persuasive to students, even if they do not follow all the
details of the argument.
Theorem 1. Let f (x) = ax with any a > 1. Then f is differentiable at x = 0 and
f
(0) > 0.
Proof. To express our geometric observations in analytic terms, let
m(x) = f (x) − f (0)/x − 0
= ax − 1/x
.
In Figure 1, x > 0 and
slope AB = m(x),
slope AC = m(−x).
We shall prove that, as x → 0 with x > 0 and x decreasing, m(x) and m(−x) approach
a common limit, which is f
(0).
To begin with, m(x) is continuous because ax is continuous. The crux of the proof,
and the only tricky part, is to show that
m(x) is increasing on (0,∞) and (−∞, 0).
We give the proof only for (0,∞) since the proof for (−∞, 0) is essentially the same.
We show first that m is increasing on the rationals in (0,∞). Fix rational numbers r
and s with 0 < r < s and let a vary with a ≥ 1. Define
g(a) = m(s) − m(r ) =as -1/s = ar-1/r
.
Then g(a) is continuous for a ≥ 1 and
g
(a) = as -1-ar-1>0 for a>0
Thus, g(a) increases as a increases and g(a) > g(1) = 0 for a > 1, so
m(r) < m(s) for 0 < r < s.
Thus, m(x) is continuous on (0,∞) and m(x) increases on the rational numbers in
(0,∞). As noted earlier, this implies that m(x) increases on (0,∞). The argument for
the interval (−∞, 0) is similar.
For x > 0,
m(−x) = m(x)a−x ,
0 < m(−x) < m(x),
0 < m(x) − m(−x) = m(x) 1 − a−x .
Let x → 0 with x decreasing. Then
m(x) decreases, m(−x) increases, m(x) − m(−x) → 0.
It follows that m(x) and m(−x) approach a common limit as x → 0, which is f
(0).
Furthermore, 0 < m(−x) < f '(0) < m(x) for x > 0, which implies that f ' (0) > 0.
We believe that this proof is new. We have been unable to find any other proof
that depends only on methods of differential calculus. Theorem 1 and familiar reasoning give the principal result on the differentiability
special refrence to www.maa.org
Sunday 15 May 2016
our blog created for always best result for others.
Give queries for better result.join this site.
Click here
Give queries for better result.join this site.
Click here
Tuesday 10 May 2016
What is function
What Are Functions?
Functions are what we use to describe things we want to talk about mathematically. I find, though, that I get a bit tongue tied when I try to define them.
The simplest definition is: a function is a bunch of ordered pairs of things (in our case the things will be numbers, but they can be otherwise), with the property that the first members of the pairs are all different from one another.
Thus, here is an example of a function:
Thus, here is an example of a function:
[{1, 1}, {2, 1}, {3, 2}]
This function consists of three pairs, whose first members are 1, 2 and 3.
It is customary to give functions names, like f, g or h, and if we call this function f, we generally use the following notation to describe it:
It is customary to give functions names, like f, g or h, and if we call this function f, we generally use the following notation to describe it:
f(1) = 1, f(2) = 1, f(3) = 2
The first members of the pairs are called arguments and the whole set of them is called the domain of the function. Thus the arguments of f here are 1, 2 and 3, and the set consisting of these three numbers is its domain.
The second members of the pairs are called the values of the functions, and the set of these is called the range of the function.
The standard terminology for describing this function f is:
The value of f at argument 1 is 1, its value at argument 2 is 1, and its value at argument 3 is 2, which we write as f(1) = 1, f(2) = 1, f(3) = 2.
We generally think of a function as a set of assignments of values (second members of our pairs) to arguments (their first members).
The condition that the first members of the pairs are all different is the condition that each argument in the domain of f is assigned aunique value in its range by any function.
Above Consider the function g, defined by the pairs (1, 1), (2, 5), (3, 1) and (4, 2). What is its domain? What is the value of g at argument 3? What is g(4)?
If you stick a thermometer in your mouth, you can measure your temperature, at some particular time. You can define a function T or temperature, which assigns the temperature you measure to the time at which you remove the thermometer from your mouth. This is a typical function. Its arguments are times of measurement and its values are temperatures.
Of course your mouth has a temperature even when you don't measure it, and it has one at every instant of time and there are an infinite number of such instants.
This means that if you want to describe a function T whose value at any time t is the temperatures in your mouth at that time, you cannot really list all its pairs. There are an infinite number of possible arguments t and it would take you forever to list them.
Instead, we employ a trick to describe a function f: we generally provide a rule which allows you, the reader, to choose any argument you like in f's domain, and, by using the rule, to compute the value of your function at that argument. This rule is often called a formula for the function. The symbol x is often used to denote the argument you will select, and the formula tells you how to compute the function at that argument.
The simplest function of all, sometimes called the identity function, is the one that assigns as value the argument itself. If we denote this function as f, it obeys
f(x) = x
for x in whatever domain we choose for it. In other words, both members of its pairs are the same wherever you choose to define it.
We can get more complicated functions by giving more complicated rules, (These rules are often called formulae as we have noted already). Thus we can define functions by giving any of the following formulae among an infinity of possibilities:
These represent, respectively, 3 times x, x squared, 3 divided by x, x divided by the sum of the square of x and 1, and so on.
We can construct functions by applying the operations of addition, subtraction, multiplication and division to copies of x and numbers in any way we see fit to do so.
There are two very nice features of functions that we construct in this way, and the first applies to all functions.
We can draw a picture of a function, called its graph on a piece of graph paper, or on a spreadsheet chart or with a graphing calculator. We can do it by taking argument-value pairs of the function and describing each by a point in the plane, with x coordinate given by the argument and y coordinate given by the value for that pair.
Of course it is impossible to plot all the pairs of a function, but we can get a pretty good idea of what its graph looks like by taking perhaps a hundred evenly spaced points in any interval of interest to us. This sounds like an impossibly tedious thing to do and it used to be so, but now it is not. On a spreadsheet, the main job is to enter the function once (with its argument given by the address of some other location). That and some copying is all you have to do, and with practice it can be done in 30 seconds for a very wide variety of functions.
The second nice feature is that we can enter any function formed by adding, subtracting, multiplying, dividing and performing still another operation, on the contents of some address very easily on a spreadsheet or graphing calculator. Not only that, these devices have some other built in functions that we can use as well.
The two of these facts mean that we can actually look at any function formed by adding subtracting multiplying or dividing copies of the identity function x and other built in functions, and any number we want, and see how they behave, with very limited effort.
We will soon see that we can use the same procedure used for constructing functions to construct their derivatives as well, but that is getting ahead of the story. We can compute derivatives for most functions numerically with only a small amount of effort as well.
Ref From math.MIT.edu
Subscribe to:
Posts (Atom)