Definition
Neighborhood of a real point x 0
Any open interval containing this point is called:
.
Here ε 1
and ε 2
- arbitrary positive numbers.
Epsilon - neighborhood of point x 0
is the set of points the distance from which to point x 0
less than ε:
.
A punctured neighborhood of point x 0
is the neighborhood of this point from which the point x itself is excluded 0
:
.
Neighborhoods of endpoints
At the very beginning, a definition of the neighborhood of a point was given. It is designated as . But you can explicitly indicate that the neighborhood depends on two numbers using the appropriate arguments:
(1)
.
That is, a neighborhood is a set of points belonging to an open interval.
Equating ε 1
to ε 2
, we get epsilon - neighborhood:
(2)
.
An epsilon neighborhood is a set of points belonging to an open interval with equidistant ends.
Of course, the letter epsilon can be replaced by any other and consider δ - neighborhood, σ - neighborhood, etc.
In limit theory, one can use a definition of neighborhood based on both set (1) and set (2). Using any of these neighborhoods gives equivalent results (see). But definition (2) is simpler, so epsilon is often used - the neighborhood of a point determined from (2).
The concepts of left-sided, right-sided and punctured neighborhoods are also widely used. endpoints. Here are their definitions.
Left neighborhood of a real point x 0
is a half-open interval located on real axis to the left of point x 0
, including the point itself:
;
.
Right-sided neighborhood of a real point x 0
is a half-open interval located to the right of point x 0
, including the point itself:
;
.
Punctured neighborhoods of endpoints
Punctured neighborhoods of point x 0 - these are the same neighborhoods from which the point itself is excluded. They are indicated with a circle above the letter. Here are their definitions.
Punctured neighborhood of point x 0
:
.
Punctured epsilon - neighborhood of point x 0
:
;
.
Pierced left side neighborhood:
;
.
Punctured right side vicinity:
;
.
Neighborhoods of points at infinity
Along with end points, neighborhoods infinitely are also introduced remote points. They are all punctured because there is no real number at infinity (the point at infinity is defined as the limit of an infinitely large sequence).
.
;
;
.
It was possible to determine the neighborhoods of points at infinity like this:
.
But instead of M, we use , so that the neighborhood with smaller ε is a subset of the neighborhood with larger ε, as for endpoint neighborhoods.
Neighborhood property
Next, we use the obvious property of the neighborhood of a point (finite or at infinity). It lies in the fact that the neighborhoods of points with smaller valuesε are subsets of neighborhoods with large values of ε. Here are more strict formulations.
Let there be a final or infinitely distant point. And let it be.
Then
;
;
;
;
;
;
;
.
The converse is also true.
Equivalence of definitions of the limit of a function according to Cauchy
Now we will show that in determining the limit of a function according to Cauchy, you can use both an arbitrary neighborhood and a neighborhood with equidistant ends.
Theorem
Cauchy definitions of the limit of a function that use arbitrary neighborhoods and neighborhoods with equidistant ends are equivalent.
Proof
Let's formulate first definition of the limit of a function.
A number a is the limit of a function at a point (finite or at infinity), if for any positive numbers there are numbers depending on and that for all belongs to the corresponding neighborhood of the point a:
.
Let's formulate second definition of the limit of a function.
The number a is the limit of the function at the point if for any positive number there is a number depending on that for all:
.
Proof 1 ⇒ 2
Let us prove that if a number a is the limit of a function by the 1st definition, then it is also a limit by the 2nd definition.
Let the first definition be satisfied. This means that there are functions and , so for any positive numbers the following holds:
at , where .
Since the numbers are arbitrary, we equate them:
.
Then there are such functions and , so for any the following holds:
at , where .
Note that .
Let be the smallest of the positive numbers and . Then, according to what was noted above,
.
If, then.
That is, we found such a function, so for any the following holds:
at , where .
This means that the number a is the limit of the function by the second definition.
Proof 2 ⇒ 1
Let us prove that if a number a is the limit of a function by the 2nd definition, then it is also a limit by the 1st definition.
Let the second definition be satisfied. Let's take two positive numbers and . And let it be the least of them. Then, according to the second definition, there is such a function , so that for any positive number and for all , it follows that
.
But according to , . Therefore, from what follows that
.
Then for any positive numbers and , we found two numbers, so for all :
.
This means that the number a is a limit by the first definition.
The theorem is proven.
Used literature:
L.D. Kudryavtsev. Well mathematical analysis. Volume 1. Moscow, 2003.
Definition. Point at infinity complex plane called isolated singular point unambiguous analytical functionf(z), If outside circle of some radius R,
those. for , there is no finite singular point of the function f(z).
To study the function at a point at infinity, we make the replacement 
Function
will have a singularity at the point ζ
= 0, and this point will be isolated, since
inside the circle 
There are no other special points according to the condition. Being analytical in this
circle (except for so-called ζ
= 0), function 
can be expanded in a Laurent series in powers ζ
. The classification described in the previous paragraph remains completely unchanged.
However, if we return to the original variable z, then series in positive and negative powers z'switch' places. Those. The classification of points at infinity will look like this:
Examples. 1.
. Dot z =
i
− pole of the 3rd order.
2.
. Dot z =
∞
− significantly singular point.
§18. Residue of an analytic function at an isolated singular point.
Let the point z 0 is an isolated singular point of a single-valued analytic function
f(z) . According to the previous, in the vicinity of this point f(z) can be represented uniquely by the Laurent series: 
Where 
Definition.Deduction analytical function f(z) at an isolated singular point z 0
called complex number, equal to the value of the integral 
, taken in the positive direction along any closed contour lying in the domain of analyticity of the function and containing within itself a single singular point z 0 .
The deduction is indicated by the symbol Res [f(z),z 0 ].
It is easy to see that the residue is at a regular or removable singular point equal to zero.
At a pole or essentially singular point, the residue is equal to the coefficient With-1 row Laurent:
.
Example. Find the residue of a function 
.
(Let it be easy to see that
coefficient With-1 is obtained when multiplying the terms with n= 0:Res[ f(z),i
]
=
}
It is often possible to calculate residues of functions over in a simple way. Let the function f(z) has incl. z 0 pole of the first order. In this case, the expansion of the function in a Laurent series has the form (§16):. Let's multiply this equality by (z−z 0) and go to the limit at 
. As a result we get: Res[ f(z),z 0 ] =
So, in
In the last example we have Res[ f(z),i
]
=
.
To calculate residues at higher order poles, multiply the function
on 
(m− pole order) and differentiate the resulting series ( m −
1) times.
In this case we have: Res[ f(z),z 0 ]
Example. Find the residue of a function 
at point z= −1.
{
Res[ f(z),
−1]
}
We defined the neighborhood of this point as the exterior of circles centered at the origin: U
(∞, ε
) = {z
∈ | |z
| > ε). Dot z
= ∞ is an isolated singular point of the analytic function w
= f
(z
), if in some neighborhood of this point there are no other singular points of this function. To determine the type of this singular point, we make a change of variable, and the point z
= ∞ goes to the point z
1 = 0, function w
= f
(z
) will take the form 
. Type of singular point z
= ∞ functions w
= f
(z
) we will call the type of singular point z
1 = 0 functions w
= φ
(z
1). If the expansion of the function w
= f
(z
) by degrees z
in the vicinity of a point z
= ∞, i.e. at sufficiently large modulus values z
, has the form , then, replacing z
on , we will receive . Thus, with such a change of variable, the main and regular parts of the Laurent series change places, and the type of the singular point z
= ∞ is determined by the number of terms in the correct part of the expansion of the function in the Laurent series in powers z
in the vicinity of a point z
= 0. Therefore
1. Point z
= ∞ is a removable singular point if this expansion does not contain the correct part (except, perhaps, for the term A
0);
2. Point z
= ∞ - pole n
-th order if the right part ends with a term A n
· z n
;
3. Point z
= ∞ is an essentially singular point if the regular part contains infinitely many terms.
In this case, the criteria for the types of singular points by value remain valid: if z= ∞ is a removable singular point, then this limit exists and is finite if z= ∞ is a pole, then this limit is infinite if z= ∞ is an essentially singular point, then this limit does not exist (neither finite nor infinite).
Examples: 1. f
(z
) = -5 + 3z
2 - z
6. The function is already a polynomial in powers z
, the highest degree is the sixth, therefore z
The same result can be obtained in another way. We will replace z
on, then 
. For function φ
(z
1) point z
1 = 0 is a pole of sixth order, therefore for f
(z
) point z
= ∞ - pole of the sixth order.
2. . For this function, obtain a power expansion z
difficult, so let's find: 
; the limit exists and is finite, so the point z
3. . Correct part of the power expansion z
contains infinitely many terms, so z
= ∞ is an essentially singular point. Otherwise, this fact can be established based on the fact that it does not exist.
Residue of a function at an infinitely distant singular point.
For the final singular point a
, Where γ
- a circuit containing no others except a
, singular points, traversed in such a way that the area bounded by it and containing the singular point remains on the left (counterclockwise).
Let's define in a similar way: 
, where Γ − is the contour limiting such a neighborhood U
(∞, r
) points z
= ∞, which does not contain other singular points, and is traversable so that this neighborhood remains on the left (i.e., clockwise). Thus, all other (final) singular points of the function must be located inside the contour Γ − . Let's change the direction of traversing the contour Γ − : 
. By the main theorem on residues 
, where the summation is carried out over all finite singular points. Therefore, finally
,
those. residue at an infinitely distant singular point equal to the sum residues over all finite singular points, taken with the opposite sign.
As a consequence, there is total sum theorem: if function w = f (z ) is analytic everywhere in the plane WITH , except for a finite number of singular points z 1 , z 2 , z 3 , …,z k , then the sum of residues at all finite singular points and the residue at infinity is zero.
Note that if z
= ∞ is a removable singular point, then the residue at it can be different from zero. So for the function, obviously, ; z
= 0 is the only finite singular point of this function, so 
, despite the fact that, i.e. z
= ∞ is a removable singular point.
If some sequence converges to finite number a , then they write
.
Previously we introduced into consideration infinitely long sequences. We assumed that they were convergent and denoted their limits with the symbols and . These symbols represent points at infinity. They don't belong to the multitude real numbers. But the concept of limit allows us to introduce such points and provides a tool for studying their properties using real numbers.
Definition
Point at infinity, or unsigned infinity, is the limit to which an infinitely large sequence tends.
Point at infinity plus infinity, is the limit to which an infinitely large sequence with positive terms tends.
Point at infinity minus infinity, is the limit to which an infinitely large sequence with negative terms tends.
For any real number a the following inequalities hold:
;
.
Using real numbers, we introduced the concept neighborhood of a point at infinity.
The neighborhood of a point is the set.
Finally, the neighborhood of a point is the set.
Here M is an arbitrary, arbitrarily large real number.
Thus, we have expanded the set of real numbers by introducing new elements into it. In this regard, there is following definition:
Extended number line or extended set of real numbers is the set of real numbers complemented by the elements and :
.
First, we will write down the properties that the points and . Next we consider the issue of strict mathematical definition operations for these points and proofs of these properties.
Properties of points at infinity
Sum and difference.
;
;
;
;
Product and quotient.
;
;
;
;
;
;
;
.
Relationship with real numbers.
Let a be an arbitrary real number. Then
;
;
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;
.
Let a > 0
. Then
;
;
.
Let a < 0
. Then
;
.
Undefined operations.
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;
;
;
;
;
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Proofs of the properties of points at infinity
Defining Mathematical Operations
We have already given definitions for points at infinity. Now we need to define mathematical operations for them. Since we defined these points using sequences, operations with these points should also be defined using sequences.
So, sum of two points
c = a + b,
belonging to the extended set of real numbers,
,
we will call the limit
,
where and are arbitrary sequences having limits
And .
The operations of subtraction, multiplication and division are defined in a similar way. Only, in the case of division, the elements in the denominator of the fraction should not be equal to zero.
Then the difference of two points:
- this is the limit: .
Product of points:
- this is the limit: .
Private:
- this is the limit: .
Here and are arbitrary sequences whose limits are a and b , respectively. IN the latter case, .
Proofs of properties
To prove the properties of points at infinity, we need to use the properties of infinitely large sequences.
Consider the property:
.
To prove it, we must show that
,
In other words, we need to prove that the sum of two sequences that converge to plus infinity converges to plus infinity.
1
the following inequalities are satisfied:
;
.
Then for and we have:
.
Let's put it. Then
at ,
Where .
This means that .
Other properties can be proved in a similar way. As an example, let's give another proof.
Let us prove that:
.
To do this we must show that
,
where and are arbitrary sequences, with limits and .
That is, we need to prove that the product of two infinitely large sequences is infinite big sequence.
Let's prove it. Since and , then there are some functions and , so for any positive number M 1
the following inequalities are satisfied:
;
.
Then for and we have:
.
Let's put it. Then
at ,
Where .
This means that .
Undefined operations
Part mathematical operations with points at infinity are not defined. To show their uncertainty, it is necessary to give a couple of special cases when the result of the operation depends on the choice of the sequences included in them.
Consider this operation:
.
It is easy to show that if and , then the limit of the sum of sequences depends on the choice of sequences and .
Indeed, let's take it. The limits of these sequences are . Amount limit
equals infinity.
Now let's take . The limits of these sequences are also equal. But the limit of their amount
equal to zero.
That is, provided that and , the value of the amount limit can take different meanings. Therefore the operation is not defined.
In a similar way, you can show the uncertainty of the remaining operations presented above.
