Definition. The singular point of the function is called isolated, if in some neighborhood of this point is an analytic function (that is, analytic in the ring).
The classification of isolated singular points of a function is related to the behavior of this function in a neighborhood of a singular point.
Definition. The point is called disposable a singular point of a function if there is a finite limit of this function at .
Example 5 Show that the function has a removable singularity at a point.
Solution. Recalling the first remarkable limit, we calculate
This means that the given function has a removable singularity at the point.
Task 4. Show that the point is removable for .
Definition. The point is called pole function , if this function increases indefinitely for , that is .
Let us pay attention to the connection between the concepts of zero and pole of an analytic function. Let's represent the function as .
If a point is a simple zero of a function, then the function has a simple pole
If the point is the order zero for the function, then for the function it is the pole order.
Example 6 Show that the function has a third-order pole at a point.
Solution. Assuming , we get . As we tend to zero, according to any law, we have . Then , and with it the function itself increases indefinitely. Therefore, , that is, the singular point is a pole. For a function, this point is obviously a triple zero. Hence, for this function, the point is a pole of the third order.
Task 5. Show that the point has a simple pole.
Definition. The point is called essentially special point of the function if at this point there is neither a finite nor an infinite limit of the function (the behavior of the function is not defined).
Let be an essential singular point of the function . Then for any preassigned complex number there is such a sequence of points converging to , along which the values tend to : ( Sochocki's theorem).
Example 7 Show that a function at a point has an essential singularity.
Solution. Consider the behavior of a given function in the vicinity of the point . For along the positive part of the real axis (i.e. ) we have and ; if along the negative part of the real axis (i.e.), then and . So there is no limit for . By definition, a function has an essential singularity at a point.
Let us consider the behavior of the function at zero from the point of view of the Sochocki theorem. Let be any complex number other than zero and infinity.
From equality we find . Assuming , we obtain a sequence of points , . Obviously, . At each point of this sequence, the function is equal to , and therefore
Task 6. Show that the function has an essential singularity at a point.
A point at infinity is always considered special for the function. A point is called an isolated singular point of a function if this function has no other singular points outside some circle centered at the origin.
The classification of isolated singular points can also be extended to the case .
Example 8 Show that the function has a double pole at infinity.
Solution. Consider the function , where is an analytic function in a neighborhood of the point , and . This means that the function has a double zero at infinity, but then for the function the point is a double pole.
Example 9 Show that the function has an essential singularity at infinity.
Solution. A similar problem is considered in pr.7. Consider the behavior of a function in the neighborhood of an infinitely distant point. For along the positive part of the real axis, and for along the negative part of the real axis. This means that there is no limit of the function at a point and, by virtue of the definition, this point is essentially singular.
The nature of the singularity of a function at a point can be judged from main part Laurent expansion in a neighborhood of this point.
Theorem 1. For the point to be disposable singular point of the function , it is necessary and sufficient that the corresponding Laurent expansion did not contain the main part.
Task 6. Using the Taylor expansion of the function in a neighborhood of the point , show that it has a removable singularity at zero.
Theorem 2. For the point to be pole functions , is necessary and sufficient so that main part corresponding Laurent expansion contained a finite number of members :
The number of the highest negative term determines the order of the pole.
In this case, the function can be represented as
where is the function analytic at the point, , is the order of the pole.
Example 10 Show that the function has simple poles at points.
Solution. Let's consider a point. We use the Laurent expansion of this function in the vicinity of this point, obtained in Example 2:
Since the highest (and only) negative power in the main part of this expansion is equal to one, the point is a simple pole of this function.
This result could have been obtained in another way. Let us represent in the form and put - this is a function that is analytic at the point and . Hence, due to (8) this function has a simple pole at the point.
Another way: consider a function that has a simple zero at the point. Hence, at this point it has a simple pole.
Similarly, if we write the function in the form , where is a function analytic at the point and , then it is immediately clear that the point is a simple pole of the function .
Task 7. Show that the function has a pole of the 2nd order at the point and a pole of the 4th order at the point .
Theorem 3. For the point to be essentially special point of the function , it is necessary and sufficient that main part Laurent expansion in a neighborhood of the point contained an infinite number of members .
Example 11. Determine the nature of the singularity at the point of the function
Solution. In the well-known expansion of the cosine, we put instead of:
Hence, the Laurent expansion in a neighborhood of a point has the form
Here the correct part is one term. And the main part contains an infinite number of terms, so the point is essentially singular.
Task 8. Show that at a point the function has an essential singularity.
Consider some function and write down its Laurent expansion at the point :
Let's make a replacement , while the point goes to the point . Now, in a neighborhood of a point at infinity, we have
It remains to introduce a new designation . We get
where is the main part, and is the regular part of the Laurent expansion of the function in the neighborhood of an infinitely distant point. Thus, in the Laurent expansion of a function in a neighborhood of a point, the main part is a series in positive powers, while the correct part is a series in negative powers. Taking into account this
However, the above criteria for determining the nature of the singularity remain valid for an infinitely distant point.
Example 12. Find out the nature of the singularity of the function at the point. , then at a point it may turn out to be non-isolated.
Example 15 The function at an infinitely distant point has an essential singularity. Show that the point for the function is not an isolated singular point.
Solution. The function has an infinite number of poles at the zeros of the denominator, that is, at the points , . Since , then the point , in any neighborhood of which there are poles , is the limit point for the poles.
Taylor series serve as an effective tool for studying functions that are analytic in the circle zol To study functions that are analytic in an annular region, it turns out that it is possible to construct expansions in positive and negative powers (z - zq) of the form that generalizes Taylor expansions. The series (1), understood as the sum of two series, is called the Laurent series. It is clear that the region of convergence of series (1) is the common part of the regions of convergence of each of the series (2). Let's find her. The area of convergence of the first series is a circle whose radius is determined by the Cauchy-Hadamard formula Inside the circle of convergence, series (3) converges to an analytic function, and in any circle of smaller radius, it converges absolutely and uniformly. The second series is a power series with respect to the variable. The series (5) converges within its circle of convergence to the analytic function of the complex variable m-*oo, and in any circle of smaller radius it converges absolutely and uniformly, which means that the region of convergence of the series (4) is the appearance of the circle - If then there is a common region of convergence of the series (3) and (4) - a circular ring in which the series (1) converges to an analytic function. Moreover, in any ring, it converges absolutely and uniformly. Example 1. Determine the region of convergence of the rad Laurent series Isolated singular points and their classification (z), which is single-valued and apolitical in a circular ring, can be represented in this ring as the sum of a convergent series whose coefficients Cn are uniquely determined and calculated by the formulas where 7p is a circle of radius m Let us fix an arbitrary point z inside the ring R We construct circles with centers at the point r whose radii satisfy the inequalities and consider a new ring. According to the Cauchy integral theorem for a multiply connected domain, we have For all points £ along the circle 7d*, the relation de the sum of a uniformly convergent series 1 1 is satisfied. Therefore, the fraction ^ can be represented in vi- /" / In a somewhat different way, for all points ξ on the circle ir> we have the relation Therefore, the fraction ^ can be represented as the sum of a uniformly convergent series in formulas (10) and (12) are analytic functions in a circular ring. Therefore, by Cauchy's theorem, the values of the corresponding integrals do not change if the circles 7/r and 7r/ are replaced by any circle. This allows us to combine formulas (10) and (12). Replacing the integrals on the right side of formula (8) with their expressions (9) and (11), respectively, we obtain the desired expansion. Since z is an arbitrary point of the ring, it follows that the series ( 14) converges to the function f(z) everywhere in this ring, and in any ring the series converges to this function absolutely and uniformly. Let us now prove that the decomposition of the form (6) is unique. Assume that one more decomposition takes place. Then, everywhere inside the ring R, we have On the circumference, the series (15) converge uniformly. Multiply both sides of the equality (where m is a fixed integer, and integrate both series term by term. As a result, we get on the left side, and on the right - Csh. Thus, (4, \u003d St. Since m is an arbitrary number, then the last equality series (6), whose coefficients are calculated by formulas (7), is called the Laurent series of the function f(z) in the ring 7) for the coefficients of the Laurent series are rarely used in practice, because, as a rule, they require cumbersome calculations. Usually, if possible, ready-made Taylor expansions of elementary functions are used. Based on the uniqueness of the expansion, any legitimate method leads to the same result. Example 2 Consider the Laurent series expansions of functions of different domains, assuming that Fuiscius /(r) has two singular points: Therefore, there are three ring domains and, centered at the point r = 0. in each of which the function f(r) is analytic: a) the circle is the circle's exterior (Fig. 27). Let us find the Laurent expansions of the function /(z) in each of these regions. We represent /(z) as a sum of elementary fractions a) Circle Transform relation (16) as follows Using the formula for the sum of terms of a geometric progression, we obtain b) The ring for the function -z remains convergent in this ring, since Series (19) for the function j^j for |z| > 1 diverges. Therefore, we transform the function /(z) as follows: applying formula (19) again, we obtain that This series converges for. Substituting the expansions (18) and (21) into relation (20), we obtain c) The exteriority of the circle for the function -z with |z| > 2 diverges, and series (21) for the function Let us represent the function /(z) in the following form: /<*> Using formulas (18) and (19), we obtain OR 1 This example shows that for the same function f(z) the Laurent expansion, generally speaking, has a different form for different rings. Example 3. Find the decomposition of the 8 Laurent series of the function Laurent series Isolated singular points and their classification in the annular region A We use the representation of the function f (z) in the following form: and transform the second term Using the formula for the sum of the terms of a geometric progression, we obtain Substituting the found expressions into the formula (22), we have Example 4. Expand the function in a Laurent series in the neighborhood of thin zq = 0. For any complex one, we have Let This expansion is valid for any point z Ф 0. In this case, the annular region is the entire complex plane with one thrown out point z - 0. This region can be defined by the following relationship: This function is analytic in the region From formulas (13) for the coefficients of the Laurent series, by the same reasoning as in the previous paragraph, one can obtain the Kouiw inequalities. if the function f(z) is bounded on a circle, where M is a constant), then isolated singular points A point zo is called an isolated singular point of the function f(z) if there exists an annular neighborhood of the point (this set is sometimes also called a punctured neighborhood of the point 2o), in where the function f(z) is single-valued and analytic. At the point zo itself, the function is either not defined or is not single-valued and analytic. Three types of singular points are distinguished depending on the behavior of the function /(z) when approaching the point zo. An isolated singular point is said to be: 1) removable if there exists a finite 2) pmusach if 3) an essentially singular point if the function f(z) has no limit for Theorem 16. An isolated singular point z0 of a function f(z) is a removable singular point if and only if the Laurent expansion of the function f(z) in a neighborhood of the point zo does not contain a principal part, i.e., has the form Let zo - removable singular point. Then there exists a finite function, hence, the function f(z) is bounded in a procological neighborhood of the point r. We set By virtue of the Cauchy inequalities Since it is possible to choose p as arbitrarily small, then all the coefficients at negative powers (z - 20) are equal to zero: Conversely, let the Laurent the expansion of the function /(r) in a neighborhood of the point zq contains only the correct part, i.e., it has the form (23) and, therefore, is Taylor. It is easy to see that for z -* z0 the function /(r) has a limit value: Theorem 17. An isolated singular point zq of the function f(z) is removable if and only if the function J(z) is bounded in some punctured neighborhood of the point zq, Zgmechai not. Let r0 be a removable singular point of f(r). Assuming we get that the function f(r) is analytic in some circle centered at the point th. This defines the name of the point - disposable. Theorem 18. An isolated singular point zq of a function f(z) is a pole if and only if the main part of the Laurent expansion of the function f(z) in a neighborhood of the point contains a finite (and positive) number of non-zero terms, i.e., has the form 4 Let z0 be a pole. Since then there exists a punctured neighborhood of the point z0 in which the function f(z) is analytic and nonzero. Then an analytic function is defined in this neighborhood and Hence, the point zq is a removable singular point (zero) of the function or where h(z) is an analytic function, h(z0) ∩ 0. is analytic in a neighborhood of the point zq, and hence, whence we obtain that Let us now assume that the function f(z) has a decomposition of the form (24) in a punctured neighborhood of the point zo. This means that in this neighborhood the function f(z) is analytic together with the function. For the function g(z), the expansion is valid from which it is clear that zq is a removable singular point of the function g(z) and exists Then the function tends at 0 - the pole of the function There is one more simple fact. The point Zq is a pole of the function f(z) if and only if the function g(z) = y can be extended to an analytic function in a neighborhood of the point zq by setting g(z0) = 0. The order of the pole of the function f(z) is called the order of zero of function jfa. Theorems 16 and 18 imply the following assertion. Theorem 19. An isolated singular thin is essentially singular if and only if the principal part of the Laurent expansion in a punctured neighborhood of this point contains infinitely many nonzero terms. Example 5. The singular point of the function is zo = 0. We have Laurent Series Isolated singular points and their classification Therefore, zo = 0 is a removable singular point. The expansion of the function /(z) in a Laurent series in the vicinity of the zero point contains only the correct part: Example7. f(z) = The singular point of the function f(z) is zq = 0. Consider the behavior of this function on the real and imaginary axes: on the real axis at x 0, on the imaginary axis Therefore, neither finite nor infinite limit f(z) at z -* 0 does not exist. Hence the point r0 = 0 is an essentially singular point of the function f(z). Let us find the Laurent expansion of the function f(z) in a neighborhood of the zero point. For any complex C we have We set. Then the Laurent expansion contains an infinite number of terms with negative powers of z.
Basic concepts and definitions:
The zero of the analytic function f(z) is the point “a” for which f(a)=0.
The zero of order “n” of the function f(z) is the point “a” if but fn(a)¹0.
A singular point "a" is called an isolated singular point of the function f(z) if there exists a neighborhood of this point where there are no singular points other than "a".
Isolated singular points are of three types: .
1 removable special points;
3 essential singular points.
The type of a singular point can be determined based on the behavior of a given function at the found singular point, as well as from the form of the Laurent series obtained for the function in the neighborhood of the found singular point.
Determining the type of a singular point by the behavior of the function in it.
1. Removable Singular Points.
An isolated singular point a of the function f(z) is called removable if there exists a finite limit .
2. Poles.
An isolated singular point a of the function f(z) is called a pole if 
.
3. Significant singular points.
An isolated singular point a of a function f(z) is called an essential singular point if neither finite nor infinite exists.
The following relation takes place between the zeros and poles of the function.
For a point a to be a pole of order n of the function f(Z), it is necessary and sufficient that this point be a zero of order n for the function .
If n=1 the pole is called simple.
Definition: An isolated singular point of a single-valued character is called:
a) removable if the main part of the decomposition is absent;
b) a pole if the main part contains a finite number of members;
c) an essentially singular point if the main part contains an infinite number of terms.
a) Thus, in a neighborhood of a removable singular point, the expansion has the form:
it expresses the function at all points of the circle |z-a| At the center z=a, the equality is false, because the function at z=a has a discontinuity, and the right side is continuous. If the value of the function in the center is changed, taking it equal to the value of the right side, then the gap will be eliminated - hence the name - removable. b) In the neighborhood of a pole of order m, the Laurent series expansion has the form: c) In the neighborhood of a simple pole Deductions and formulas for their calculation. The residue of an analytic function f(z) at an isolated singular point z 0 is a complex number equal to the value of the integral The residue of the function f(z) at an isolated singular point z 0 is denoted by the symbol Res f(z 0) or Res (f(z); z 0). In this way, Resf(z0)= If we put n=-1 in the formula (22.15.1), then we get: C-1= or Res f(z 0)= C -1 , those. the residue of the function f(z) with respect to the singular point z 0 is equal to the coefficient of the first term with a negative exponent in the expansion of the function f(z) in a Laurent series. Calculation of deductions. Regular or removable singular points. Obviously, if z=z 0 is a regular or removable singular point of the function f(z), then Res f(z 0)=0 (there is no principal part in the Laurent decomposition in these cases, so c-1=0). Pole. Let the point z 0 be a simple pole of the function f(z). Then the Laurent series for the function f(z) in a neighborhood of the point z 0 has the form: From here Therefore, passing in this equality to the limit as z --z 0 , we obtain Res f(z0)= Essentially special point. If the point z 0 is an essentially singular point of the function f(z), then to calculate the residue of the function at this point, one usually directly determines the coefficient c-1 in the expansion of the function in a Laurent series. Event classification. Sum, product of events, their properties, graphical representation. Events are divided into: 1. Random 2. Credible 3. Impossible Reliable - this is an event that necessarily occurs in these conditions (night is followed by morning). Random is an event that may or may not occur (passing an exam). The impossible is an event that will not occur under the given conditions (get a green pencil out of the box with only red ones).
, taken in the positive direction along the circle L centered at the point z 0 , which lies in the region of analyticity of the function f(z) (i.e., in the ring 0<|z-z0| . (22.15.1)
