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71
a
!
!3!
2
n
!
n
(0)(0)(0)(0)...
26
++++
1...
1!2!
=+++
x-1
!
!
2
!
1
n
n
R1lim
=
-
n
¥®
a
n
Example. Find the interval of convergence of a series
32
x
n
xxx
...
...
+++++
Find the radius of convergence
1
a
-
n
1
R
n
==
¥®
a
n
-
n
)!1(
limlim
=
1
n
!
lim
n
!
lim
-
n
nnn
)!1(
¥==
n
¥®¥®¥®
.
Hence, the given series converges for any value of X. The term of
n
thise series tends to zero.
x
lim =
¥®
n
.0
Decomposition of functions in power series is of great importance for
solving different problems of the study of functions, differentiation, integration, solution of differential equations, computation of limits, calculation of the approximate function values.
The idea of matching derivatives by powers is becoming central to
this lecture. The derivatives are given at a base point (say x = 0). They
are numbers f(0), f'(0)…
The Taylor series that matches f(x) and all its derivatives at x = 0 is
ffхfxfx
¢¢¢¢¢¢
11
23
The new function is ех. All its derivatives are ех. At x = 0, this function and its derivatives equal 1. To match these l's, we move factorials
into the denominators. Term by term the series is
Example. Be decomposed into a number of function
If you apply to the same function formula Taylor
¢
)0(
f
)0()(
fxf
+=
x
+
we get:
2
)0(
1
n
+++
xx
.
)(
xRx
n
,
x
e
)(
¢¢
)0(
f
2
x
n
f
...
72
¢
×
xf
¢¢
xf
¢¢¢
xf
1
)(
=
-
x
2
)(
=
)1(
-
x
32
)(
=
)1(
-
x
¢
;1)0(;
=
f
2
)1(
¢¢
f
3
;2)0(;
=
¢¢¢
;!3)0(;
=
f
4
……………………………….
!
)(
n
xf
n
)(
=
n
+
)1(
x
-
In the end, we get:
)(
n
1
;!)0(;
nf
=
2
n
......1)(
+++++=
xxxxf
Reading questions
1. What are the applications of the decomposition of a function into a
power series?
2. That matches the Taylor series?
3. Under what conditions functional series is called convergent?
73
Section 9. Elements of the complex
variable function theory
Topic 24. Complex variable function theory
Introduction. In this lecture we will present some elements of the
theory of functions of a complex variable. It should be noted that this
theory of course transcends the limits of this lecture and the student can
deepen their knowledge on the topic using additional literature.
Complex numbers.
Definition. A complex number z is an expression z = a + ib, where a
and b are real numbers, i is an imaginary unit, defined by a correlation:
i2 = –1; i =
The number a is called a real part of z (a = Re z), and b - the imagi-
nary part (b = Im z).
If a =Re z =0, then z is imaginary, if b = Im z = 0, then z is real.
Definition. Numbers z = a + ib and z = a + ib are called complex
conjugate.
Definition. Two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2 are
equal, if their real and imaginary parts are equal:
a1 = a2; b1 = b2.
Definition. A complex number is equal to zero, if it’s real and imaginary parts are respectively equal to zero: a = b = 0
The concept of complex numbers has a geometric interpretation. If
any real number can be geometrically represented as points on the number line, then the complex number is represented by a point on the plane,
the coordinates of which are respectively real and imaginary parts of a
complex number. The horizontal axis will be the real numeric axis, and
the vertical – the imaginary axis.
Thus, the real numbers are on the ОХ axis , and the imaginary numbers are on the ОY axis.
Using such geometric representations we can represent numbers in
the so-called trigonometric form.
.1-
74
Figure 9
j=j
=
j+j=j+j=+
=
a
±+±=+±+=±
=
Fig.9 – a geometric interpretation of complex numbers .
The trigonometric form of a number.
In terms of geometry, we can see that
sin;cos rbra
the complex number can be represented as:
)sin(cossincos
irirribaz
This form is called a trigonometric form of a complex number.
The value r is called a modulus of a complex number, and the angle
j – the argument of a complex number.
zArgzr =j= ;
.
From a geometrical point of view it is seen:
22
b
arctgzArgbaibar ==j+=+=
;;
Obviously, complex-conjugate numbers have the same moduli and
opposite arguments.
.; zArgzArgzz -==
Actions with complex numbers.
Basic actions with complex numbers are derived from actions with
polynomials.
1) Adding and subtracting.
)()()()(
bbiaaibaibazzz
2121221121
2
21
2
)()( bbaaz ±+±=
21
2) Multiplying.
))(( bbiaibbiaaaibaibazzz +++=++==
abbaibbaazzz ++-==
2
212121221121
)()(
2121212121
21
In trigonometric form:
)sin(cos
j+j= irz
1111
,
).sin(cos
j+j= irz
2222
))sin()(cos(
j+j+j+j== irrzzz
21212121
. Then
75
In case of complex – conjugate numbers:
+
-
+
22
22
.))((
zzbaibaibazz ==+=-+=
3) Dividing.
z
1
z +=
z
2
ibaiba
-+
z
=
z
2211
ibaiba
-+
2222
=
2
+
2
iba
==
))((
11
iba
+
22
=
))((
bbaa
2121
i
+
2
ba
2
iyx
)()(
babaibbaa
-++
2
2
ba
+
2
2
baba
2112
2
2
ba
+
2
2
21122121
In trigonometric form:
r
z
1
1
z
r
z
2
2
))sin()(cos(
j-j+j-j== i
2121
4) Powering.
From the multiplication of complex numbers, it follows that:
In the general case we get:
22
nn
)2sin2(cos
j+j== irzzz
.
.
)sin(cos j+j= ninrz
Where n is a positive integer.
This expression is called the de Moivre formula. (Abraham de Moivre
(1667 – 1754) – English mathematician)
De Moivre's formula can be used to find the trigonometric functions
of double, triple, etc. angles.
5) Taking the root of a complex number.
n
n
By powering, we get:
Thus:
n
n
n
Hence, the root of n
n
.;2; Zkknr
Îp+j=y=r
p+j
k
æ
n
=j+j=
rirz
ç
th
power of a complex number has n different
è
2
cos)sin(cos
n
)sin(cos)sin(cos y+yr=j+j= iirz
.
.
)sin(cos)sin(cos j+j=y+yr irnin
p+j
k
2
+
i
sin
ö
÷
n
ø
values.
The exponential form of a complex number
Let’s consider this exponential function
z
.; iyxzew
+==
We can say, that function w can be written this way:
xiyx
+
+==
)sin(cos yiyeew
.
76
j=i
+==
=
[
]
±=±
[
]
×=×
!
!2!
1
n
This equation is called the Euler equation:
rez
.
Definition. If every complex number z from a set D according to
some rule set in accordance with a certain complex number w from the
set G, then this field is set to a function of a complex variable representing the set D to the set G.
w = f(z)
A complex function can be written in the form:
),(),()( yxivyxuzfw
;
)(Re),(
zfyxu
)(Im),(
=
zfyxv
;
u, v – there are a valid functions from variables х and у.
Properties of functions of a complex variable.
For functions of a complex variable f(z) and g(z) have the following
properties:
1)
2)
3)
lim
®
)(lim
)(
zf
zz
0
)(
zg
zf
®
zz
0
)(lim
zg
®
zz
0
®
zz
0
)(lim)(lim)()(lim
zgzfzgzf
zzzzzz ®®®
000
)(lim)(lim)()(lim
zgzfzgzf
zzzzzz ®®®
000
.0)(lim;
¹=
zg
Consider the decomposition into a power series of the following func-
tions:
sin
z
1
e
z
1cos
z
2
!5!3!1
!4!2
n
zzz
...
-+-+-=
...
+++++=
1253
+
n
zzzz
n
)1(...
n
zzz
n
)1(...
...
+
)!12(
+
242
n
)!2(
n
...
+-+-+-=
Functions e z , cosz, sinz are connected by Euler's formula.
This formula can be very easily obtained by adding the appropriate
series.
iz
-
zize
sincos +=
.
It also performs the equality:
77
i
iziz
2
(
)
;;
zzzz
zzzz
shzeechzee
chzeeshzee
(cos1sin1)(cos1sin1)
2
i
+===
==
sin1cos12sin12cos1.
22
-
ee
+
z
cos
=
2
iziz
-
ee
-
z
sin
=
+
tgz
ctgz
zzzz
2121
sin
cos
cos
sin
z
z
==
z
==
+
xiyxiyxz
m
+===
zm
;;
eeeee ==
iziz
-
ee
-
iziz
-
)(
eei
+
iziz
-
eeizz
+
iziz
-
ee
-
)sin(cos yiyeeeee
;
2 ziz
p+
;
ee =
;
)(
;
Definition. Hyperbolic sine, cosine, tangent and cotangent functions
are called, respectively:
zzzz
--
eeee
-+
shzchz
==
thzcthz
====
;;
22
--
-+
--
+-
Example. Find sin(1+2i).
2222
iiii
----
eeeeee
sin(12)
==
=+=+
i
22
-
eiei
--
cos1()sin1()
eeiee
2222
--
eeee
+-
--
22
ii
+--
2222
-++
2
i
ichsh
Reading questions
1. How to get Euler's formula?
2. What are the three properties of functions of complex variable?
3. What is the purpose in this lecture we used the decomposition of a
function into a power series?
78
Section 10. Basics of probability analysis
n
)
Topic 25. Probability theory
Introduction. We will review the main aspects of the theory of probability. However, a quantity of theoretical positions from this section of
mathematics will not be discussed because not enough time. In this lecture we will learn how to calculate probability using the classic definition. We learn how to calculate the probability of an event a certain
number of times.
Definition. The event is called every fact that can happen or not happen in the experiment.
Definition. Events are called incompatible if the appearance of one of
them excludes the other.
Definition. Reliable event is an event that is certain to occur as a result of experience. The event is called impossible, if it will never happen
as a result of experiment.
Definition. The probability of the event A is called a mathematical
evaluation of the possibility of the occurrence as a result of experi-
m
ence
favorable event to A outcomes of the experiment to the total number of
pair wise incompatible outcomes of experiment.
the other white. Find the probability that removed at random ball will be
red, green or white.
of green as the event B, the appearance of white as the event C.
compatible events is equal to the sum of the probabilities of these events
Р(А + В) = Р(А) + Р(В).
Р(А) + Р(
AP =)(
.
The probability of the event A is equal to the ratio of the number of
0 ≤ P(A) ≤ 1
Example. In the box there are 10 balls. 3 of them are red, 2 – green,
We denote the appearance of a red ball as an event A, the appearance
Then, in accordance with recorded above formulas we get:
3
)( === CPBPAP
10
2
)(;
10
5
;
)(;
10
Theorem (sum of probabilities). The probability of the sum of two in-
The sum of the probability of the opposite events is equal to one.
A
) = 1.
79
Definition. The probability, calculated assuming that was an event A,
!1!
4
!2!
3
=++
=
is called the conditional probability of event B:
РА(В) = Р(В / А) = Р(АВ) / Р(А).
If the result of the experiment may receive the n events are independent in the combination, then the probability of occurrence of at least one
of them is calculated by the formula: Р(А) = 1 – q1q2…qn.
Repeated tests. The Bernoulli Formula.
If you do a number of tests, which may happen or not to happen event
A and the probability of occurrence of this event in each of the tests does
not depend on results of other tests, such tests are called independent
relative to the event A.
Suppose that event A occurs in each test with a probability of Р(А)=р.
Find the probability Pm,n, resulting in n tests event A came exactly m
times.
The solution to this problem is implemented in the Bernoulli formula
P
= )1(
nm
,
!
mnm
-
)!(!
mnm
-
pp
-
.
n
The formula for Bernoulli important fact that is true for any number
of independent tests.
Example. Target is 5 shots. The probability of hit for each shot equal
to 0.4. Find the probability that the target hit at least three times.
Because the shots are independent, we can apply the formula for the
Bernoulli probability that in n tests event probability p occurs exactly m
times.
P
= )1(
nm
,
Five hits out of five:
5,5
Four hits out of five shots:
Three hits out of five:
5,3
!
mnm
-
)!(!
55
01024,04,0
=== pP
!5
= ppP
5,4
!5
= ppP
×
4
×
23
=-
mnm
-
pp
-
0768,0)1(
=-
2304,0)1(
n
Finally, we obtain the probability of not less than three hits out of five
shots:
P
31744,02304,00768,001204,0
Reading questions
1. That allows you to calculate the Bernoulli formula?
2. What is the sum of the probability of the opposite events?
3. The probability of my hitting the target equal to 0.3. What is the
80
Topic 26. Elements of mathematical statistics
Age of students, X
17 18 19 20 21
The number of st
u
dents,
f 3 5 7 4 2
Introduction. In this lecture we will discuss the most important concepts of mathematical statistics and learn how to calculate the variance
of two ways.
In science the term statistics introduced by the German scientist Gottfried Achenwall in 1746. He proposed to replace the name of the course
"political science", which was in universities in Germany, on "Statistics".
He put the beginning of the development of statistics as a science and
academic discipline.
Any study statistics event has as common for the whole entirety and
specific, individual properties. The difference between individual phenomena is called variation. Average values express the proximity of the
characteristics of individual phenomena.
The formula for the arithmetic mean value has the form:
The formula is called the weighted arithmetic mean.
Example. Distribution of full-time students by age
The average age must be a result of the uniform distribution of the total age of all students.
Total age of all students, according to initial information, can be obtained as the sum of the characteristic values in each group Xi on the
number of students with the same age f (frequency).
Have the formula:
N
fХ
ii
å
=
i
1
=
X
å
i
N
=
1
,
f
i
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