CAUCHY’S THEOREM FOR A MULTIPLY CONNECTED DOMAIN.If a function f z is analytic in a multiply connected domain D bounded by a contour Γ consisting of several closed curves and is continuo
Trang 1If f (z) and g(z) are analytic functions in a simply connected domain D and z = a and
z = b are arbitrary points of the domain D, then the formula of integration by parts holds:
b
a f (z) dg(z) = f (b)g(b) – f (a)g(a) –
b
If an analytic function z = g(w) determines a single-valued mapping of a curve 2 C onto
C f (z) dz =
2
C f (g(w))g
CAUCHY’S THEOREM FOR A SIMPLY CONNECTED DOMAIN.If a function f (z) is analytic
in a simply connected domain D bounded by a contour C and is continuous in D, then
7
C f (z) dz =0
CAUCHY’S THEOREM FOR A MULTIPLY CONNECTED DOMAIN.If a function f (z) is analytic
in a multiply connected domain D bounded by a contour Γ consisting of several closed
curves and is continuous in D, then7
Γf (z) dz = 0provided that the sense of all curves formingΓ is chosen in such a way that the domain D lies to one side of the contour.
If a function f (z) is analytic in an n-connected domain D and continuous in D, and
C is the boundary of D, then for any interior point z of this domain the Cauchy integral
formula holds:
f (z) = 1
2πi
C
f (ξ)
(Here integration is in the positive sense of C; i.e., the domain D lies to the left of C.)
Under the same assumptions as above, formula (10.1.2.26) implies expressions for the value
of the derivative of arbitrary order of the function f (z) at any interior point z of the domain:
f(n)
z (z) = 2n πi!
C
f (ξ) (ξ – z) n+1 dξ (n =1,2, ) (10.1.2.27)
For an arbitrary smooth curve C, not necessarily closed, and for a function f (ξ) every-where continuous on C, possibly except for finitely many points at which this function has
an integrable discontinuity, the right-hand side of formula (10.1.2.26) defines a Cauchy-type
integral The function F (z) determined by a Cauchy-type integral is analytic at any point
that does not belong to C If C divides the plane into several domains, then the Cauchy-type
integral generally determines different analytic functions in these domains
Formulas (10.1.2.26) and (10.1.2.27) allow one to calculate the integrals
C
f (ξ)
ξ – z dξ=2πif (z),
C
f (ξ) (ξ – z) n dξ=
2πi
n! f
(n)
Example 4 Let us calculate the integral
C
Im z dz, where C is the semicircle|z| = 1 , 0 ≤arg z≤π(Fig 10.4).
X
1 1
Y
C
Figure 10.4 The semicircle|z| = 1 , 0 ≤arg z≤π.
Trang 2Using formula (10.1.2.21), we obtain
C
Im z dz =
C
y (dx + i dy) =
C
y dx + i
C
y dy=
–1
1
√
1– x2dx – i
–1
1
x dx= –π
2.
Example 5 Let us calculate the integral
C
dz
z – z0, where C is the circle of radius R centered at a point z0with anticlockwise sense.
Using the Cauchy integral formula (10.1.2.28), we obtain
C
1
z – z0 dz= 2πi.
Example 6 Let us calculate the integral
C
dz
z2+ 1,
where C is the circle of unit radius centered at the point i with anticlockwise sense.
To apply the Cauchy integral formula (10.1.2.26), we transform the integrand as follows:
1
(z – i)(z + i) =
1
z + i
1
z – i =
f (z)
z + i, f (z) =
1
z + i. The function f (z) =1/ (z + i) is analytic in the interior of the domain under study and on its boundary; hence
the Cauchy integral formula (10.1.2.26) and the first of formulas (10.1.2.28) hold From the latter formula, we
C
dz
z2+ 1 =
C
f (z)
z – i dz=2πif (i) =2πi1
2i = π.
Formulas (10.1.2.26) and (10.1.2.27) imply the Cauchy inequalities
|f(n)
z (z)| ≤ 2n!
π
C
f (ξ) (ξ – z) n+1 dξ
≤ 2n !M l
where M = max
z Df (z)is the maximum modulus of the function f (z) in the domain D, R
is the distance from the point z to the boundary C, and l is the length of the boundary C.
If, in particular, f (z) is analytic in the disk D =|z – z0|< R, and bounded in ¯ D, then we obtain the inequality
|f(n)
z (z0)| ≤ n !M
R n (n =0,1,2, ) (10.1.2.30)
MORERA’S THEOREM If a function f (z) is continuous in a simply connected domain D
and7
C f (z) dz =0for any closed curve C lying in D, then f (z) is analytic in the domain D.
10.1.2-5 Taylor and Laurent series
If a series
∞
n=0
of analytic functions in a simply connected domain D converges uniformly in this domain, then its sum is analytic in the domain D.
If a series (10.1.2.31) of functions analytic in a domain D and continuous in D converges uniformly in D, then it can be differentiated termwise any number of times and can be integrated termwise over any piecewise smooth curve C lying in D.
Trang 3ABEL’S THEOREM If the power series
∞
n=0
converges at a point z0, then it also converges at any point z satisfying the condition
|z – a| <|z0– a| Moreover, the series converges uniformly in any disk|z – a| < q| z0– a|,
where0< q <1
It follows from Abel’s theorem that the domain of convergence of a power series is an
open disk centered at the point a; moreover, this disk can fill the entire plane The radius of this disk is called the radius of convergence of a power series The sum of the power series
inside the disk of convergence is an analytic function
Remark. The radius of convergence R can be found by the Cauchy–Hadamard formula
1
R = lim
n→∞
n
|c n| , where lim denotes the upper limit.
If a function f (z) is analytic in the open disk D of radius R centered at a point z = a, then this function can be represented in this disk by its Taylor series
f (z) =
∞
n=0
c n (z – a) n,
whose coefficients are determined by the formulas
c n= f
(n)
z (a)
n! =
1
2πi
C
f (ξ) (ξ – z) n+1 dξ (n =0,1,2, ), (10.1.2.33)
where C is the circle |z – a| = qR, 0 < q < 1 In any closed domain belonging to the
disk D, the Taylor series converges uniformly Any power series expansion of an analytic
function is its Taylor expansion The Taylor series expansions of the functions given in
Paragraph 10.1.2-3 in powers of z have the form
e z =1+ z + z
2
2! +
z3
cos z =1– z2
2! +
z4
4! – , sin z = z –
z3
3! +
z5
5! – (|z|<∞), (10.1.2.35)
cosh z =1+ z
2
2! +
z4
4! + , sinh z = z +
z3
3! +
z5
5! + (|z|<∞), (10.1.2.36) ln(1+ z) = z – z
2
2! +
z3
(1+ z) a=1+ az + a (a –1)
2! z2+
a (a –1)(a –2) 3! z3+ (|z|<1). (10.1.2.38) The last two expansions are valid for the single-valued branches for which the values of the
functions for z =0are equal to0and1, respectively
Trang 4Remark Series expansions (10.1.2.34)–(10.1.2.38) coincide with analogous expansions of the corre-sponding elementary functions of the real variable (see Paragraph 8.3.2-3).
To obtain the Taylor series for other branches of the multi-valued function Ln(1+ z),
one has to add the numbers2kπi , k = 1, 2, to the expression in the right-hand side:
Ln(1+ z) = z – z2
2! +
z3
3! – +2kπ. The domain of convergence of the function series ∞
n=–∞ c n (z – a)
nis a circular annulus
K : r <|z – a| < R, where0 ≤r ≤∞ and0 ≤R≤∞ The sum of the series is an analytic
function in the annulus of convergence Conversely, in any annulus K where the function
f (z) is analytic, this function can be represented by the Laurent series expansion
f (z) =
∞
n=–∞
c n (z – a) n
with coefficients determined by the formulas
c n= 1
2πi
γ
f (ξ) (ξ – z) n+1 dξ (n =0, 1, 2, ), (10.1.2.39)
where γ is the circle|z – a| = ρ, r < ρ < R In any closed domain contained in the annulus K,
the Laurent series converges uniformly
The part of the Laurent series with negative numbers,
– 1
n=–∞
c n (z – a) n=
∞
n=1
c–n
is called its principal part, and the part with nonnegative numbers,
∞
n=0
is called the regular part Any expansion of an analytic function in positive and negative powers of z – a is its Laurent expansion.
Example 7 Let us consider Laurent series expansions of the function
f (z) = 1
z( 1– z)
in a Laurent series in the domain 0 < |z| < 1 This function is analytic in the annulus 0 < |z| < 1 and hence can
be expanded in the corresponding Laurent series We write this function as the sum of elementary fractions:
f (z) = 1
z( 1– z) =
1
z + 1
1– z.
Since |z| < 1 , we can use formula (10.1.2.39) and obtain the expansion
1
z( 1– z) =
1
z + 1+ z + z2+
Example 8 Let us consider Laurent series expansions of the function
f (z) = e1/z
in a Laurent series in a neighborhood of the point z0 = 0 To this end, we use the well-known
expan-sion (10.1.2.34), where we should replace z by1/z Thus we obtain
e1/z= 1 + 1
1!z +
1
2!z2 + +
1
n !z n + (z≠ 0 ).
Trang 510.1.2-6 Zeros and isolated singularities of analytic functions.
A point z = a is called a zero of a function f (z) if f (a) = 0 If f (z) is analytic at the
point a and is not zero identically, then the least order of nonzero coefficients in the Taylor expansion of f (z) centered at a, in other words, the number n of the first nonzero derivative
f(n) (a), is called the order of zero of this function In a neighborhood of a zero a of order n,
the Taylor expansion of f (z) has the form
f (z) = c n (z – a) n + c n+1(z – a) n+1+ (c n≠ 0, n≥ 1)
In this case, f (z) = c n (z – a) n g (z), where the function g(z) is analytic at the point a and
g (a)≠ 0 A first-order zero is said to be simple The point z =∞ is a zero of order n for a
function f (z) if z =0is a zero of order n for F (z) = f (1 /z)
If a function f (z) is analytic at a point a and is not identically zero in any neighborhood
of a, then there exists a neighborhood of a in which f (z) does not have any zeros other than a.
UNIQUENESS THEOREM If functions f (z) and g(z) are analytic in a domain D and their values coincide on some sequence a k of points converging to an interior point a of the domain D, then f (z)≡g (z) everywhere in D.
ROUCH E´’S THEOREM If functions f (z) and g(z) are analytic in a simply connected domain D bounded by a curve C, are continuous in D, and satisfy the inequality|f (z)|>|g (z)|
on C, then the functions f (z) and f (z) + g(z) have the same number of zeros in D.
A point a is called an isolated singularity of a single-valued analytic function f (z) if there exists a neighborhood of this point in which f (z) is analytic everywhere except for the point a itself The point a is called
1 A removable singularity if lim
z→a f (z) exists and is finite.
2 A pole if lim
z→a f (z) = ∞.
3 An essential singularity if lim
z→a f (z) does not exist.
A necessary and sufficient condition for a point a to be a removable singularity of a function f (z) is that the Laurent expansion of f (z) around a does not contain the principal part If a function f (z) is bounded in a neighborhood of an isolated singularity a, then a is
a removable singularity of this function
A necessary and sufficient condition for a point a to be a pole of a function f (z) is that the principal part of the Laurent expansion of f (z) around a contains finitely many terms:
f (z) = c–n (z – a) n + +
c–1
(z – a) +
∞
k=0
The order of a pole a of a function f (z) is defined to be the order of the zero of the function F (z) = f (1 /z ) If c–n≠ 0in expansion (10.1.2.42), then the order of the pole a of the function f (z) is equal to n For n =1, we have a simple pole
A necessary and sufficient condition for a point a to be an essential singularity of a function f (z) is that the principal part of the Laurent expansion of f (z) around a contains
infinitely many nonzero terms
SOKHOTSKII’S THEOREM If a is an essential singularity of a function f (z), then for each complex number A there exists a sequence of points z k → a such that f(z k)→ A.
Trang 6Example 9 Let us consider some functions with different singular points.
1◦ The function f (z) = (1– cos z)/z2has a removable singularity at the origin, since its Laurent expansion about the origin,
1– cos z
z2 = 1
2 –
z2
24 +
z4
720 – ,
does not contain the principal part.
2◦ The function f (z) =1/( 1+e z2) has infinitely many poles at the points z = √
( 2k+ 1)πi (k =0 , 1 , 2, ).
All these poles are simple poles, since the function 1/f (z) =1+ e z2has simple zeros at these points (Its derivative is nonzero at these points.)
3◦ The function f (z) = sin(1/z) has an essential singularity at the origin, since the principal part of its Laurent expansion
sin 1
z = 1
z – 1
z33 ! + contains infinitely many terms.
The following two simplest classes of single-valued analytic functions are distinguished according to the character of singular points
1 Entire functions A function f (z) is said to be entire if it does not have singular points
in the finite part of the plane An entire function can be represented by an everywhere convergent power series
f (z) =
∞
n=0
c n z n.
An entire function can have only one singular point at z = ∞ If this singularity is a pole of
order n, then f (z) is a polynomial of degree n If z = ∞ is an essential singularity, then f(z)
is called an entire transcendental function If z = ∞ is a regular point (i.e., f(z) is analytic
for all z), then f (z) is constant (Liouville’s theorem) All polynomials, the exponential function, sin z, cos z, etc are examples of entire functions Sums, differences, and products
of entire functions are themselves entire functions
2 Meromorphic functions A function f (z) is said to be meromorphic if it does not
have any singularities except for poles The number of these poles in each finite closed
domain D is always finite.
Suppose that a function f (z) is analytic in a neighborhood of the point at infinity The
definition of singular points can be generalized to this function without any changes But the criteria for the type of a singular point at infinity related to the Laurent expansion are different
THEOREM In the case of a removable singularity at the point at infinity, the Laurent
expansion of a function f (z) in a neighborhood of this point does not contain positive powers of z In the case of a pole, it contains finitely many positive powers of z In the case
of an essential singularity, it contains infinitely many powers of z.
Let f (z) be a multi-valued function defined in a neighborhood D of a point z = a except possibly for the point a itself, and let f1(z), f2(z) be its branches, which are single-valued continuous functions in the domain where they are defined The point a is called a branch
point (ramification point) of the function f (z) if f (z) passes from one branch to another
as the point z goes along a closed curve around the point z in a neighborhood of D If the original branch is reached again for the first time after going around this curve m times (in the same sense), then the number m –1is called the order of the branch point, and the point a itself is called a branch point of order m –1
Trang 7If all branches f k (z) tend to the same finite or infinite limit as z → a, then the point a is
called an algebraic branch point (For example, the point z =0is an algebraic branch point
of the function f (z) = m √
z.) In this case, the single-valued function
F (z) = f (z m + a) has a regular point or a pole for z =0
If the limit of f k (z) as z → a does not exist, then the point a is called a transcendental
branch point For example, the point z =0is a transcendental branch point of the function
f (z) = exp( m
1/z)
In a neighborhood of a branch point a of finite order, the function f (z) can be expanded
in a fractional power series (Puiseux series)
f (z) =
∞
k=–∞
If a new branch is obtained each time after going around this curve (in the same sense),
then the point a is called a branch point of infinite order (a logarithmic branch point) For example, the points z =0and z = ∞ are logarithmic branch points of the multivalued
function w = Ln z A logarithmic branch point is classified as a transcendental branch point For a ≠ ∞, the expansion (10.1.2.43) contains finitely many terms with negative k
(infinitely many in the case of a transcendental point)
10.1.2-7 Residues
The residue res f (a) of a function f (z) at an isolated singularity a is defined as the number
res f (a) = 1
2πi
8
where the integral is taken in the positive sense over a contour C surrounding the point a and containing no other singularities of f (z) in the interior.
Remark. Residues are sometimes denoted by res[f (z); a] or res z=a f (z).
The residue res f (a) of a function f (z) at a singularity a is equal to the coefficient of (z – a)–1in the Laurent expansion of f (z) in a neighborhood of the point a,
res f (a) = 1
2πi
8
Basic rules for finding the residues:
1 The residue of a function at a removable singularity is always zero
2 If a is a pole of order n, then
res f (a) = 1
(n –1)!z→alim
d n–1
dz n–1
f (z)(z – a) n
3 For a simple pole (n =1),
res f (a) = lim
z→a
f (z)(z – a) n
...f(n) (a), is called the order of zero of this function In a neighborhood of a zero a of order n,
the Taylor expansion of f (z) has the form
f (z) = c n... order of the pole a of the function f (z) is equal to n For n =1, we have a simple pole
A necessary and sufficient condition for a point a to be an essential singularity of a function... (10.1.2.38) The last two expansions are valid for the single-valued branches for which the values of the
functions for z =0are equal to 0and1 , respectively
Trang