The residue at the singular point z = ai of the function f z, which is a first-order pole and which is the only singular point of this function lying inside the contour, can be calculate
Trang 1416 FUNCTIONS OFCOMPLEXVARIABLE
4 If f (z) is the quotient of two analytic functions,
f (z) = ϕ (z)
ψ (z),
in a neighborhood of a point a and ϕ(a)≠ 0, ψ(a) =0, but ψ z (a)≠ 0(i.e., a is a simple pole
of f (z)), then
res f (a) = ϕ (a)
ψ
5 If a is an essential singularity of f (z), then to obtain res f (a), one has to find the coefficient c–1in the Laurent expansion of f (z) in a neighborhood of a.
A function f (z) is said to be continuous on the boundary C of the domain D if for each boundary point z0there exists a limit lim
z→z0f (z) = f (z0) as z → z0 , z D CAUCHY’S RESIDUE THEOREM Let f (z) be a function continuous on the boundary C
of a domain D and analytic in the interior of D everywhere except for finitely many points
a1, ,a n Then
C f (z) dz =2πin
k=1
res f (a k), (10.1.2.49)
where the integral is taken in the positive sense of C.
The logarithmic residue of a function f (z) at a point a is by definition the residue of its
logarithmic derivative
ln f (z)
z =
f
z (z)
f (z).
THEOREM The logarithmic derivative f z (z)/f (z)has first-order poles at the zeros and
poles of f (z) Moreover, the logarithmic residue of f (z) at a zero or a pole of f (z) is equal
to the order of the zero or minus the order of the pole, respectively
The residue of a function f (z) at infinity is defined as
res f ( ∞) = 2πi1
8
whereΓ is a circle of sufficiently large radius|z|= ρ and the integral is taken in the clockwise sense (so that the neighborhood of the point z = ∞ remains to the left of the contour, just
as in the case of a finite point)
The residue of f (z) at infinity is equal to minus the coefficient of z–1 in the Laurent
expansion of f (z) in a neighborhood of the point z = ∞,
Note that
res f ( ∞) = lim
provided that this limit exists
THEOREM If a function f (z) has finitely many singular points a1, ,a nin the extended complex plane, then the sum of all its residues, including the residue at infinity, is zero:
res f ( ∞) +n
k=1
res f (a k) =0 (10.1.2.53)
Trang 2Example 10 Let us calculate the integral
8
C
ln(z +2 )
z2 dz,
where C is the circle|z| = 12.
In the disk |z| ≤ 1
2, there is only one singular point of the integrand, z =0 , which is a second-order pole.
The residue of f (z) at z =0 is calculated by the formula (10.1.2.46)
res f (0 ) = lim
z→0
*
z2ln(z +2 )
z2
+
z= lim
z→0[ln(z +2 )] z= lim
z→0
1
z+ 2 =
1
2.
Using formula (10.1.2.44), we obtain
1
2 =
1 2πi
8
C
ln(z +2 )
z2 dz,
8
C
ln(z +2 )
z2 dz = πi.
10.1.2-8 Calculation of definite integrals
Suppose that we need to calculate the integral of a real function f (x) over a (finite or infinite) interval (a, b) Let us supplement the interval (a, b) with a curve Γ that, together with (a, b), bounds a domain D, and then analytically continue the function f (x) into D Then the residue theorem can be applied to this analytic continuation of f (z), and by this theorem,
b
a f (x) dx +
Γf (z) dz =2πiΛ, (10.1.2.54) whereΛ is the sum of residues of f(z) in D If7Γf (z) dz can be calculated or expressed
in terms of the desired integral7b
a f (x) dx, then the problem will be solved.
When calculating integrals of the form7∞
–∞ f (x) dx, one should apply (10.1.2.49) to the contour C that consists of the interval (–R, R) of the real axis and the arc C Rof the circle
|z| = R in the upper half-plane Sometimes it is only possible to find the limit as R → ∞
of the integral over the contour C Rrather than to calculate it, and often it turns out that the limit of this integral is equal to zero
The integral over the curveΓ can be estimated using the following lemmas
JORDAN LEMMA If a function g(z) tends to zero uniformly with respect to arg z along
a sequence of circular arcs C R n :|z|= R n , Im z > –a (where R n → ∞ and a is fixed), then
lim
n→∞
C Rn g (z)e
imz dz =0 (10.1.2.55)
for each positive number m.
If a function f (z) is analytic for|z|> R0and zf (z) →0as|z|→ ∞ for y≥ 0, then
lim
R→∞
C R
where C Ris the arc of the circle|z|= R in the upper half-plane.
Trang 3418 FUNCTIONS OFCOMPLEXVARIABLE
X R
ai
R
Y
C R
Figure 10.5 The contour to calculate the Laplace integral.
Example 11 (Laplace integral) To calculate the integral
∞
0
cos x
x2+ a2 dx, one uses the auxiliary function
f (z) = e
iz
z2+ a2 = g(z)e
iz, g (z) = 1
z2+ a2 and the contour shown in Fig 10.5 Since g(z) satisfies the inequality|g(z)|< (R2– a2)–1on C R, it follows
that this function uniformly tends to zero as R → ∞, and by the Jordan lemma we obtain
C R
f (z) dz =
C R
g (z)e iz dz →0
as R → ∞.
By the residue theorem, R
–R
e ix
x2+ a2 dx+
C R
f (z) dz =2πi2aie–a
for each R >0 (The residue at the singular point z = ai of the function f (z), which is a first-order pole and which
is the only singular point of this function lying inside the contour, can be calculated by formula (10.1.2.48).)
In the limit as R → ∞, we obtain
∞
–∞
e ix
x2+ a2 dx=
π
ae a Separating the real part and using the fact that the function is even, we obtain
∞
0
cos x
x2+ a2 dx=
π
2aea.
10.1.2-9 Analytic continuation
Let two domains D1and D2have a common part γ of the boundary, and let single-valued analytic functions f1(z) and f2(z), respectively, be given in these domains The function
f2(z) is called a direct analytic continuation of f1(z) into the domain D2 if there exists a
function f (z) analytic in the domain D1∪ γ ∪ D2and satisfying the condition
f (z) =f
1(z) for z D1,
f2(z) for z D2. (10.1.2.57)
If such a continuation is possible, then the function f (z) is uniquely determined If the domains are simply connected and the functions f1(z) and f2(z) are continuous in D1∪ γ
and D2∪ γ, respectively, and coincide on γ, then f2 (z) is the direct analytic continuation
of f1(z) into the domain D2 In addition, suppose that the domains D1and D2are allowed
to have common interior points A function f2(z) is called a direct analytic continuation
of f1(z) through γ if f1(z) and f2(z) are continuous in D1∪ γ and D2 ∪ γ, respectively,
and their values on γ coincide At the common interior points of D1and D2, the function determined by relation (10.1.2.57) can be double-valued
Trang 410.2 Main Applications
10.2.1 Conformal Mappings
10.2.1-1 Generalities
A one-to-one mapping
w = f (z) = u(x, y) + iv(x, y) (10.2.1.1)
of a domain D onto a domain D ∗ is said to be conformal if the principal linear part of this mapping at any point of D is an orthogonal orientation-preserving transformation.
Main properties of conformal mappings:
1 Circular property A conformal mapping takes infinitesimal circles to infinitesimal
circles (up to higher-order infinitesimals)
2 Angle preservation property A conformal mapping preserves the angles between
intersecting curves at points of intersection
THEOREM A function w = f (z) is a conformal mapping of a domain D if and only if
it is analytic and schlicht in D and the derivative f z (z) vanishes nowhere in D.
The main problem in the theory of conformal mappings is as follows: given domains D and D ∗, construct a function that gives a conformal mapping of one of the domains onto the other
THE MAIN THEOREM OF THE THEORY OF CONFORMAL MAPPINGS(RIEMANN THEOREM)
For any simply connected domains D and D ∗ (with boundaries consisting of more than a
single point), any points z0D and w0D ∗ , and any real number α0, there exists a unique conformal mapping
w = f (z)
of D onto D ∗such that
f (z0) = w0, arg f z (z0) = α0
10.2.1-2 Boundary correspondence
On the boundary C of a domain D, let us introduce a real arc length parameter s reckoned from some point of C, so that ζ = ζ(s) on C If f (z) is a continuous function in the closed domain D, then on the boundary C one can set
f (ζ) = f [ζ(s)] = ϕ(s).
The function ϕ(s) is called the boundary function for f (z).
THEOREM ON THE BOUNDARY CORRESPONDENCE Suppose that a function w = f (z) specifies a conformal mapping between domains D and D ∗ Then the following assertions hold
1 If the boundary of D ∗ does not have infinite branches, then f (z) is continuous on the boundary of D and the boundary function w = f (ζ) = ϕ(s) is a continuous one-to-one correspondence between the boundaries of the domains D and D ∗
2 If the boundaries of D and D ∗do not contain infinite branches and have a continuous
curvature at each point, then the boundary function ϕ(s) is continuously differentiable.
Trang 5420 FUNCTIONS OFCOMPLEXVARIABLE
BOUNDARY CORRESPONDENCE PRINCIPLE Let D and D ∗ be two simply connected
domains with boundaries C and C ∗ , and let the domain D ∗ be bounded Suppose that a
function w = f (z) satisfies the following conditions:
1 It is analytic in D and continuous in D If the point at infinity lies in the interior of the domain D ∗, then the boundary correspondence principle remains valid provided
that w = f (z) is continuous in D and analytic in D everywhere except for an interior point z0, at which this function has a simple pole
2 It is a one-to-one sense-preserving mapping of C onto C ∗
Then f(z) is a (schlicht) conformal mapping of D onto D ∗
Example 1 The exponential function w = e zmaps
a) the strip between the straight lines y = k(x – a1) and y = k(x – a2 ) onto the strip lying between the
logarithmic spirals (Fig 10.6) (If k(a2– a1 ) = 2π , then the spirals coincide, and we obtain a mapping onto
the plane with the spiral cut; for k(a2– a1 ) > 2π , the mapping is not schlicht);
b) the strip 0< Im z < π onto the upper half-plane (Fig 10.7); here the point πi is taken to the point –1 , and the point 0 is taken to the point 1 ;
c) the half-strip 0< Im z < π, Re z <0 , onto the half-disk |w| < 1, Im w >0 (Fig 10.8);
d) a rectangle onto a half-annulus (Fig 10.9).
Figure 10.6 The exponential function w = e z maps the strip between the straight lines onto the strip lying between the logarithmic spirals.
X
πi
1
Figure 10.7 The exponential function w = e zmaps the strip 0< Im z < π onto the upper half-plane.
πi
Figure 10.8 The exponential function w = e zmaps the half-strip 0< Im z < π, Re z <0 , onto the half-disk.
Trang 6X U
πi
Figure 10.9 The exponential function w = e zmaps a rectangle onto a half-annulus.
Example 2 The function w = z2maps the interior of a circle onto the interior of a cardioid (Fig 10.10).
The circle given in polar coordinates by the equation r = cos ϕ is taken to the cardioid ρ = 12( 1+ cos θ), where θ =2ϕ
X
Figure 10.10 The function w = z2maps the interior of a circle onto the interior of a cardioid.
Example 3 The function w = √
zmaps the interior of a circle onto the interior of the right branch of
a lemniscate (Fig 10.11) The circle r = cos ϕ is taken to the right branch of the lemniscate ρ = √
cos 2θ ,
where θ =12ϕ.
X
Figure 10.11 The function w = √
z maps the interior of a circle onto the interior of the right branch of a lemniscate.
Example 4 The function w = – ln(1– z) maps the interior of the unit circle onto the interior of the curve
u= – ln( 2cos v) (Fig 10.12).
Example 5 The function w = ln z–1
1+ z maps the upper half-plane onto the strip0< Im z < π (Fig 10.13) The function z = – coth12wspecifies the inverse mapping of the strip 0< Im z < π onto the upper half-plane.
Trang 7422 FUNCTIONS OFCOMPLEXVARIABLE
X O
π π
2
2
Figure 10.12 The function w = – ln(1– z) maps the interior of the unit circle onto the interior of the curve
u= – ln( 2cos v).
X
πi
Figure 10.13 The function w = ln z–1
1+ z maps the upper half-plane onto the strip0< Im z < π.
10.2.1-3 Linear-fractional mappings
The mappings given by linear-fractional functions
w= az + b
where a, b, c, and d are complex constants and ad – bc ≠ 0, are called linear-fractional mappings The function (10.2.1.2) is defined on the extended complex plane (Its value at the point z = –d/c is defined to be ∞, and the value at the point z = ∞ is defined to be a/c.)
A linear-fractional function defines a schlicht mapping of the extended z-plane onto the extended w-plane Linear-fractional functions are the only functions with this property Points z and z ∗ are said to be symmetric about the circle C0:|z– z0|= R0if they lie on
the same ray passing through z0and|z– z0||z∗ – z0|= R20
The transformation taking each point z to the point z ∗ symmetric to z about the circle C0
is called the symmetry, or the inversion, about the circle.
Points z and z ∗ are symmetric about a circle C0if and only if they are the vertices of a
pencil of circles orthogonal to the circle C0
THEOREM An arbitrary linear-fractional function
w= az + b
cz + d, ad – bc≠ 0,
defines a schlicht conformal mapping of the extended z-plane onto the extended w-plane This mapping transforms any circle on the extended z-plane into a circle on the extended
w -plane (the circular property) and transforms any pair of points symmetric about a circle C into a pair of points symmetric about the image of the circle C (preservation of symmetric
points)