Download COMPLEX ANALYSIS Contents 1. Complex numbers 1 2

COMPLEX ANALYSIS
JAN MALÝ
Contents
1. Complex numbers
2. Holomorphic functions
3. Elementary functions of complex variable
4. Integration over paths
5. Index of a point with respect to a path
6. Cauchy theorem and its consequences
7. Power series
8. Laurent series and residues
9. Complex analysis methods of interal calculus
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1. Complex numbers
1.1. Complex numbers. The complex plane C is a structure (R2 , times, R, i), it satisfies the algebraic
axioms of field http://en.wikipedia.org/wiki/Field (mathematics) The euclidean space R2 is there
with its structure of a normed linear (vector) space. The operation “times” is complex multiplication.
The field R of reals is embedded into C so that the real number a is identified with the complex number
(a, 0). i is defined as (0, 1). The multiplication obeys the following rules:
• commutative, associative, distributive w.r.t. addition,
• compatible with “old” multiplication on R,
• i2 = −1.
We write a complex number z = (z1 , z2 ) rather in the form z1 + iz2 , (z1 , z2 ∈ R). We write z1 = Re z,
z2 = Im z. If u1 , u2 , v1 , v2 are reals, the multiplication rule is
(u1 + iu2 )(v1 + iv2 ) = u1 v1 − u2 v2 + i(u1 v2 + u2 v1 ).
1.2. Rules. We define absolute value alias modulus
q
|z| = z12 + z22 ,
complex conjugate
z̄ = z1 − iz2 .
We have
(1)
1.3. Division.
uv = u v,
z̄¯ = z,
z + z̄
z − z̄
Re z =
, Im z =
,
2i
√2
|z| = z z̄.
v
= vz −1 ,
z
where
z −1 =
1
z̄
.
|z|2
1.4. Goniometric functions. We define functions cos and sin of real variable according to the rule
(cos t + i sin t)0 = − sin t + i cos t,
sin 0 = 0,
cos 0 = 1.
This leads to a system of ordinary differential equations which is known to have an unique solution.
We define π as the smallest positive solution of sin t = 0. The functions sin, cos and the constant π so
defined have all properties which we know from elementary mathematics (where however the exposition
is not rigorous).
1.5. Argument. We say that u ∈ C is a complex unit if |u| = 1. For each complex unit we find a real t
so that
u = cos t + i sin t.
More generally, each complex number can be represented as
(2)
z = r(cos t + i sin t),
where r = |z| and t belongs to the set of arguments of z. Any argument t of z has the meaning of the
→
angle of 0z with the positive real axis. If T is argument of z, then all other arguments are of the form
t = T + 2kπ, k integer. If
(3)
z ∈ D := {z ∈ C : Im z 6= 0 nebo Re z > 0}.
we find an unique argument of z in the interval (−π, π). This argument is called the principal value of
argument of z and denoted arg z. Thus arg is a real function on D. We have
z2
Re z > 0,
(4)
arg(z1 + iz2 ) = arctg ,
z1
or
z
2
,
z ∈ D.
(5)
arg z = 2 arctg
z1 + |z|
1.6. Rules. If u, v and uv ∈ D, then there exists “correction integer” k ∈ {−1, 0, 1} such that
(6)
arg(uv) = arg u + arg v − 2kπ.
Similarly, if n ∈ Z, u and un ∈ D, then there exists k ∈ Z such that
(7)
arg(un ) = n arg u − 2kπ.
Let Arg z be the set of all arguments of z, we have
(8)
u ∈ Arg a,
v ∈ Arg b =⇒ u + v ∈ Arg ab,
u ∈ Arg a,
n ∈ Z =⇒ nu ∈ Arg an .
1.7. Geometric interpretation of multiplication. If u, v are nonzero complex numbers, we express
them in the goniometric form
u = r(cos α + i sin α),
v = s(cos β + i sin β)
and observe that
uv = rs cos(α + β) + i sin(α + β)
We observe that absolute values are multiplied, whereas argumants are added.
2. Holomorphic functions
2.1. Complex derivative and holomorphic functions. Let G ⊂ C be an open set and f : G → C
be a function. We say that f is holomorphic at a point w ∈ G if there exists a limit
L = lim
z→w
f (z) − f (w)
.
z−w
This limit L is called a complex derivative of f at w and is denoted by f 0 (w). We say that f is holomorphic
in G if it is holomorphic at each point w ∈ G. If f is holomorphic at w, then f is continuous at w.
2
2.2. Cauchy-Riemann conditions. We can understand f as a R2 valued mapping f = (f1 , f2 ) on
an open set G ⊂ R2 . Then there is a correspondence between real analytic interpretation and complex
analytic. If f has a derivative f 0 as a function of complex variable, then it has also a derivative in the
sense of two real variables and all entries of the the Jacobi matrix
!
∂f1
∂f1
(z)
(z)
∂z1
∂z2
.
∂f2
∂f2
∂z1 (z)
∂z2 (z)
make sense. The converse is however invalid. If we have a differentiable R2 -valued mapping of two
variables, only under an exceptional situation this represents a holomorphic function, namely, it is
holomorphic if and only if it satisfies a system of partial differential equations as stated below.
2.3. Theorem (On Cauchy-Riemann conditions). Let G ⊂ C be an open set and f : G → C be a mapping
differentiable at a point z ∈ G. Then the function f = f1 + if2 is holomorphic at z = z1 + iz2 ∈ G, iff
∂f1
∂f2
∂f1
∂f2
(9)
(z) =
(z),
(z) = −
(z).
∂z1
∂z2
∂z2
∂z1
2.4. Observation. If a holomorphic function is only real-valued, then it is locally constant. Therefore
functions |z|, |z|2 , arg z, Re z, Im z are not holomorphic. Also the function z 7→ z̄ is not holomorphic.
2.5. Theorem (rules). Derivative of a constant function is 0. Let f, g be holomorphic functions on an
open set G ⊂ C. Then
(f + g)0 = f 0 + g 0 ,
(f g)0 = f 0 g + g 0 f,
f 0
f 0 g − g0 f
=
na {g 6= 0}.
g
g2
2.6. Theorem (derivative of a superposition). Let f be holomorphic on an open set G ⊂ C and h be
holomorphic on an open set containing f (G). Then at each z ∈ G we have
(h ◦ g)0 (z) = (h0 (f (z))g 0 (z).
2.7. Theorem (derivative of the inverse). Let f be an one-to-one holomorphic function on an open set
G ⊂ C and h is the inverse to f . Then f 0 is continuous and nonvanishing, f (G) is an open set, h is
holomorphic on f (G) and
1
h0 (f (z)) = 0 ,
z ∈ G.
f (z)
3. Elementary functions of complex variable
3.1. Integer power. For n ∈ N we define z n by repeated multiplication, z 0 = 1, z −n = 1/z n . The
negative powers are defined only outside the origin. If z = r (cos t + i sin t), then
z n = rn (cos nt + i sin nt).
The function z n is holomorphic and satisfies
(z n )0 = nz n−1 .
3.2. Exponential function. We define the exponential function exp of complex variable so that we
obtain
(exp z)0 = exp z,
exp 0 = 1.
We have
(10)
exp(x + iy) = exp x (cos y + i sin y),
x, y ∈ R.
For u, v, z ∈ C, n ∈ Z we have the rules
exp(u + v) = exp u exp v,
(exp z)n = exp nz.
The function exp has a period 2πi and nowhere attains 0. We also use the notation ez for exp z. Notice
eit = exp it = cos t + i sin t,
t ∈ R.
This allows us an easier calculation with goniometric functions based on rules
ei(s+t) = eis eit ,
eint = (eit )n .
3
3.3. Goniometric functions. We define goniometric functions of complex variable as
(11)
cos z = 12 (eiz + e−iz ),
sin z =
iz
1
2i (e
− e−iz ),
z ∈ C.
These functions are holomorphic na C and satisfy familiar rules. For the derivative we have
(cos z)0 = − sin z,
(sin z)0 = cos z.
3.4. Logarithm. The function exp is not one-to-one and thus it does not have a usual inverse. The
“complete logarithm” is a multivalued relation, not a function. For each w ∈ C \ {0} we introduce the
set Ln w of all solutions z of the equation exp z = w. The principal value of the logarithm is defined on
D (see (3)) as
ln w = ln |w| + i arg w.
Then
z = ln w if
| Im z| < π & exp z = w .
The function ln is holomorphic on D and satisfies
1
.
w
If u, v and uv ∈ D, then there exists “correction integer” k ∈ {−1, 0, 1} such that
(ln w)0 =
(12)
ln(uv) = ln u + ln v − i2kπ.
(13)
n
Similarly, if n ∈ Z, u and u ∈ D, then there exists k ∈ Z such that
(14)
ln(un ) = n ln u − i2kπ.
Recall that Ln w = {z ∈ C : exp z = w}, then we have
u ∈ Ln a, v ∈ Ln b =⇒ u + v ∈ Ln ab,
(15)
u ∈ Ln a, n ∈ Z =⇒ nu ∈ Ln an .
3.5. General power. Let a ∈ C. Then the “complete” a-th power is a multivalued relation, not a
function. For each z ∈ C \ {0} we introduce the set
Ma (z) = {exp au : u ∈ Ln z}.
For z ∈ D we define the principal value of the a-th power by
(16)
z a = exp(a ln z),
z ∈ D.
(If a ∈ Z, then formula (16) also holds but it does not have the meaning of definition.) The function z a
is holomorphic on D. We have
(z a )0 = az a−1 ,
z ∈ D.
n
For w 6= 0, the equation z = w has n solutions, which form vertices of a regular n-gon with center at 0.
If a is real from (−1, 1), then the function z a maps D onto the angle {z : | arg z| < aπ}, in particular
√
z = z 1/2 maps D onto {z : Re z > 0}. So, the square root of a number z ∈ D is a complex number
w such that Re w > 0 and w2 = z. We can compute the square root by the method of bisection of the
angle, namely
p
√
z + |z|
.
z = |z| z + |z|
For calculation with powers we must be careful, otherwise we can obtain wrong statements as
−1 = i4/2 = (i4 )1/2 = 11/2 = 1.
sin z
3.6. Arcustangens and arcussinus. The function tg z = cos
z has a period π and thus the inverse
relation Arctg is not graph of a function. The function tg is one-to-one on | Re z| < π/2 and maps this
set onto
E = {w ∈ C : Im w 6= 0 or |w| < 1}.
The principal value of arcustangens is defined on E as
1 + iw
1
arctg w =
ln
.
2i
1 − iw
We have
z = arctg w iff
| Re z| < π2 & tg z = w .
4
The function arctg is holomorphic on E and satisfies
1
.
1 + w2
Similarly, the principal value of arcussinus is defined on
(arctg w)0 =
(17)
F = {w ∈ C : Re w 6= 0 or |w| < 1}.
as
p
1
ln 1 − w2 + iw .
i
iff
| Re z| < π2 & sin z = w .
arcsin w =
We have
z = arcsin w
The function arcsin is holomorphic on F and satisfies
(arcsin w)0 = √
1
.
1 − w2
4. Integration over paths
4.1. Path, chain, cycle. A path is a piecewise C 1 -smooth mapping of a closed interval ha, bi to C. The
initial point of ϕ is ϕ(a), the end point is ϕ(b). By −̇ϕ we mark the path t 7→ ϕ(−t), t ∈ h−b, −ai; its
initial point is ϕ(b) and end point is ϕ(a), it is opposite to ϕ. A path ϕ is said to be closed if ϕ(a) = ϕ(b).
A chain is a finite sequence of paths. We use the notation
ψ = ϕ1 +̇ϕ2 +̇ . . . +̇ϕm .
The +̇ symbol is used to distinguish this linking of paths from addition of their values. A cycle is a chain
of closed paths. The geometric image of a chain ψ is defined as the set
m
[
hψi :=
ϕj (haj , bj i);
j=1
a similar notation is used for single paths. We write simply
ϕ1 −̇ϕ2
instead of
ϕ1 +̇(−̇ϕ2 ).
Consider a set G ⊂ C. We say that ψ is a chain in G if hψi ⊂ G, similarly for paths or cycles.
4.2. Curve integral. Let ϕ : ha, bi → C be a path, G ⊂ C be an open set containing hϕi and f : G → C
be a continuous function. Define
Z
Z b
f (z) dz :=
f (ϕ(t)) ϕ0 (t) dt.
ϕ
a
Integral over a chain is defined as sum of integrals over its individual paths, i.e.
Z
Z
Z
f (z) dz :=
f (z) dz + · · · +
f (z) dz.
ϕ1 +̇ϕ2 +̇...+̇ϕm
ϕ1
ϕm
Of course,
Z
Z
f (z) dz = −
f (z) dz.
−̇ϕ
ϕ
4.3. A useful estimate. The curve integral of a function f over a path ϕ can be estimated as
Z
f (z) dz ≤ `(ϕ) sup |f (z)|,
z∈hϕi
ϕ
where
Z
`(ϕ) :=
Z
ds =
ϕ
a
is the length of the path ϕ.
5
b
|ϕ0 (t)| dt
4.4. Examples. 1. Circle. The circle with center at w and radius r can be parametrized in “polar
coordinates”
ϕ(t) = w + r eit ,
t ∈ (−π, π).
If U is the open disc in C with center at w and radius r, the above path is called the oriented boundary
of U and denoted as ∂U .
2. Line segment. Let A, B ∈ C. We denote by AB the line segment ϕ from A to B, namely
ϕ(t) = A + t(B − A),
t ∈ h0, 1i.
3. Boundary of triangle. Triangle is an ordered triple T = [A, B, C] of complex numbers. We say that
a triangle T is degenerate if A, B, C are colinear, otherwise the triangle is nondegenerate. With each
triangle T = [A, B, C] we associate the triangle-set
4T = 4ABC := {λA + µB + νC : λ, µ, ν ≥ 0, λ + µ + ν = 1}.
If T is degenerated, 4T can be a line segment of a point. The path
∂T := AB +̇BC +̇CA
is the oriented boundary of T . Let G ⊂ C be a set. We say that T is triangle in G if 4T ⊂ G.
4.5. Antiderivative. Let G ⊂ C be an open set and f, g : G → C be functions. We say that f is the
antiderivative of g in G if f 0 = g in G.
4.6. Starlike set. A set G ⊂ C is called starlike, if there exists a point A ∈ G such that for each point
B ∈ G, the entire line segment AB lies in G. We also say that G is starlike with respect to A. The
sets D, E, F, which we introduced as domains for elementary functions, are all starlike. A set G ⊂ C is
convex if it is starlike with respect to any of its points.
4.7. Observation. Let f be an antiderivative of g and c is a constant, then f +c is also an antiderivative.
Conversely, if G is starlike, all antiderivatives of g on G differ only by additive constants.
4.8. Theorem (On antiderivative). Let G ⊂ C be an open set and f, g : G → C be continuous functions.
The following conditions are equiavalent:
(i) f is an antiderivative of g.
(ii) For each path ϕ : ha, bi → G we have
Z
f (ϕ(b)) − f (ϕ(a)) =
g(z) dz.
ϕ
(iii) for each line segment AB in G we have
Z
f (B) − f (A) =
g(z) dz.
AB
4.9. Theorem (On existence of an antiderivative). Let G ⊂ C be an open set and g : G → C be a
continuous function. Consider the following conditions:
(i) g has an antiderivative in G.
(ii) For each closed path ϕ in G we have
Z
g(z) dz = 0,
ϕ
(iii) For each triangle T in G we have
Z
g(z) dz = 0.
∂T
Then
(a) (i) =⇒ (ii) =⇒ (iii),
(b) For G starlike, all the three conditions are equivalent.
4.10. Exercise. Verify the Cauchy-Riemann conditions for the functions exp, ln. Show that the CauchyRiemann conditions are violated for the function z̄ (evaluation of the complex conjugate).
√
4.11. Exercise. Show that f (z) = 1 + z 2 is well defined in the unit disc U = {z ∈ C : |z| < 1} and
compute f 0 (z).
q
1+z
4.12. Exercise. Show that f (z) = ln 1−z
is well defined in the unit disc U and compute f 0 (z).
6
4.13. Exercise. Show that the function
√
z cannot be continuously extended to the entire complex plane.
4.14. Exercise. Verify that (17) and (12) are compatible with Theorem 2.7.
4.15. Exercise. Prove that ln(−z) is an antiderivative of
1
z
on {z ∈ C : − z ∈ D}.
4.16. Exercise. Compute
Z
dz
,
∂U z
where U = {z ∈ C : |z| < 1} is the unit disc. (Use the polar coordinates.)
4.17. Exercise. Compute
Z
dz
,
AB z
where A = [1, 0], B = [0, 1]. (Use that the logarithmic function is the antiderivative of z1 .)
4.18. Exercise. Compute
Z
dz
,
AB +̇BC z
where A = [1, 0], B = [0, 1], C = [−1, 0]. (Use 4.17 and 4.15.)
7
5. Index of a point with respect to a path
5.1. Increment of argument. Let ϕ : ha, bi → C be a path in C \ {0}. Then there exists a continuous
function Aϕ : ha, bi → R such that for each t ∈ ha, bi the value Aϕ (t) is an argument (not necesarilly
principal value) of ϕ(t):
Aϕ (t) ∈ Arg(ϕ(t)),
t ∈ ha, bi.
The number Aϕ (b) − Aϕ (a) is called the increment of the argument along the path ϕ. The function
Aϕ is determined by the path ϕ up to a additive constant, which for the calculation of the increment is
immaterial. The increment of the argument along a chain is defined as sum of increments of its individual
paths.
5.2. Index of a point with respect to a path. Let w ∈ C and ψ be a closed path, or more generally
a cycle, in C \ {w}. Denote by ψ − w the translated path
(ψ − w)(t) = ψ(t) − w.
As a consequence of closedness of ψ we obtain that the increment of the argument along ψ − w is an
integer multiple of 2π. We define the index of w with respect to ψ, denoted by indψ w, as this integer m
such that the increment of the argument along ψ − w is 2πm.
If ψ is a closed path (or, more generally, a cycle), we denote
Int ψ = {z ∈ C \ hψi : indψ z 6= 0},
Ext ψ = {z ∈ C \ hψi : indψ z = 0}.
5.3. Theorem (on index). Let w ∈ C and ψ be a cycle in C \ {w}. Then
Z
dz
1
.
indψ w =
2πi ψ z − w
5.4. Jordan curve. A curve is a continuous mapping ϕ of a closed interval ha, bi to C. (It is like a path,
but without the piecewise smoothness assumption.) The curve ϕ is closed if ϕ(a) = ϕ(b). We can define
index also for curves.
A closed curve which does not intersect itself is called a Jordan path. (The term “does not intersects”
means that ϕ is injective on ha, b)). The so called J ordan theorem says that for each Jordan curve ϕ,
the sets Int ϕ and Ext ϕ are nonempty and connected. The Jordan theorem looks intuitively clear, but
its proof is difficult.
6. Cauchy theorem and its consequences
6.1. Theorem (Cauchy theorem for a triangle). Let G ⊂ C be an open set, T be a triangle in G, and
f : G → C be a holomorphic function. Then
Z
f (z) dz = 0.
∂T
6.2. Theorem (Cauchy theorem). Let G ⊂ C be an open set, ϕ be a cycle, hϕi ∪ Int ϕ ⊂ G, and
f : G → C be a holomorphic function. Then
Z
f (z) dz = 0.
ϕ
6.3. Theorem (Cauchy formula). Let G ⊂ C be an open set, ϕ be a cycle, hϕi∪Int ϕ ⊂ G and f : G → C
be a holomorphic function. Let w ∈ G, indϕ w = 1. Then
Z
1
f (z)
f (w) =
dz.
2πi ϕ z − w
6.4. Theorem (Cauchy formula for derivatives). Let G ⊂ C be an open set, and f : G → C be holomorphic function. Then f has in G (complex) derivatives of all orders. Let ϕ be a cycle, hϕi ∪ Int ϕ ⊂ G.
Let w ∈ G, indϕ w = 1. Then
Z
1
f (z)
f 0 (w) =
dz,
2πi ϕ (z − w)2
generally,
Z
k!
f (z)
f (k) (w) =
dz.
2πi ϕ (z − w)k+1
8
6.5. Corollary. Let G ⊂ C be an open set and f : G → C has an antiderivative in G. Then f is
holomorphic v G.
6.6. Theorem (Morera). Let G ⊂ C be an open set and f : G → C be a continuous function. If
Z
f (z) dz = 0,
∂T
for each triangle T in G, then f is holomorphic in G.
6.7. Theorem (Liouville). Let f : C → C be holomorphic and bounded. Then f is constant.
6.8. Theorem (Fundamental theorem of algebra). Let f : C → C be a nonconstant polynomial. Then f
has a root in C, this means, there exists a solution w of the equation f (w) = 0.
7. Power series
7.1. Power series. A power series is a series in the form
∞
X
(18)
an (z − w)n ,
n=0
where w ∈ C is the center of the power series and an are coefficients. The radius of convergence of the
power series (18) is


if (18) converges only at w,
0
R= ∞
if (18) converges everywhere,


radius of the largest disc U (w, r) in which (18) converges otherwise.
Recall that
U (w, r) := z ∈ C : |z − w| < r .
The change of variables (translation) z 0 = z − w simplifies the investigation of general power series to
that of power series centered at the origin. Thus, we will primarily study the series
∞
X
(19)
an z n ,
n=0
The examination of convergence of power series follows the same methods as in the real variable
theory.
7.2. Theorem (Special Dirichlet criterion). Let {an } be a sequence of real numbers, an & 0. Then the
series (19) converges for all complex numbers z with |z| ≤ 1, with the only possible exception at z = 1.
(The criterion cannot help to resolve the problem of convergence at 1.)
7.3. Examples. For the series
∞
X
(20)
(−1)k
k=1
z 2k
k
2
we make a change of variables v = −z , which leads to
(21)
∞
X
vk
k=1
k
.
The series (21) converges by the Dirichlet criterion for each complex number v satisfying |v| ≤ 1, except
v = 1 (where it diverges as we easily observe). Thus the initial series (20) converges, iff |z| ≤ 1 and
z 2 6= −1, this means for z 6= ±i ; at the points z = ±i it diverges.
The series
∞
X
zk
k2
k=1
converges by the Dirichlet criterion for each complex number v with |v| ≤ 1, except for at v = 1, where
the series diverges as well, but now not according to the Dirichlet criterion.
9
7.4. Theorem (Differentiability of the sum of a power series). Let {an } be a sequence of complex numbers
and u, w ∈ C. Let the series
∞
X
(22)
an (z − w)n
n=0
converge at a point z = u. Then the series (22) converges in the disc
U = {z ∈ C : |z − w| < |u − w|},
and its sum f is holomorphic in U . Namely, we have
∞
X
(23)
f 0 (z) =
nan (z − w)n−1 .
n=0
7.5. Theorem (Abel). Let {an } be a sequence of complex numbers and u ∈ C. Let the series
(24)
f (z) =
∞
X
an z n
n=0
converges at z = u. Then
f (u) = lim f (ru).
r→1−
7.6. Theorem (on the radius of convergence). Let {an } be a sequence of complex numbers and w ∈ C.
Let R be the radius of convergence of
∞
X
an (z − w)n .
n=0
Suppose that there exists a limit
λ := lim |an |1/n .
(25)
n→∞
Then
1
.
λ
→ ∞, then we have R = 0.
R=
If |an |1/n → 0, then we have R = +∞, if |an |1/n
If there exists a limit
|an+1 |
,
|an |
then there exists also the limit (25) and µ = λ. In this case we can compute r as 1/µ.
µ := lim
n→∞
7.7. Theorem (On the Taylor expansion). Let f be holomorphic in the disc U = {z ∈ C : |z − w| < r}.
Then there exist complex numbers an such that
∞
X
an (z − w)n ,
z ∈ U.
f (z) =
n=0
The coefficients an are
f (n) (w)
.
n!
7.8. Taylor series. The series from Theorem (7.7) is called the Taylor series of the function f with
center at w. We can evaluate the coefficients from the formula (26) or by the method of differentiation
term by term using (23).
(26)
an =
7.9. Exampley. The Taylor series of the function exp z is
∞
X
zn
exp z =
,
z ∈ C,
n!
n=1
this is easily seen from (26). Taylor series of functions sin and cos is
sin z =
∞
X
k=0
(−1)k
z 2k+1
,
(2k + 1)!
cos z =
∞
X
(−1)k
k=0
10
z 2k
,
(2k)!
z ∈ C,
it is derived using Euler’s rules (11). Taylor series of the function ln(1 + z) is
(27)
ln(1 + z) =
∞
X
(−1)n−1
n=1
zn
,
n
|z| ≤ 1, z 6= −1.
Indeed, by geometric series expansion we have
∞
∞
X
X
1
=
(−1)n−1 z n−1 =
(−1)k z k .
1 + z n=1
k=0
and integrating both parts we obtain (27). Similarly we derive (make carefully as an exercise!!)
arctg z =
∞
X
(−1)k
k=0
z 2k+1
,
2k + 1
|z| ≤ 1, z 6= ±i.
The binomial series
(28)
a
(1 + z) =
∞ X
a
n=0
n
zn,
|z| < 1,
can be derived from (26). Recall that binomial coefficients are defined also for noninteger, even complex
a as
a
a(a − 1) . . . (a − n + 1)
.
=
n!
n
If a = 0, 1, 2, . . . then the series (28) has only finite number of summands and thus it converges in C.
Otherwise its radius of convergence is 1. The convergence of the binomial series at points of the unit circle
depends on the exponent a and the discussion, if required with proofs, is difficult. If |z| = 1, z 6= −1,
then (28) converges at z if Re a > −1 and diverges otherwise. If z = −1, then then (28) converges at z
if Re a > 0 or a = 0 and diverges otherwise.
The Taylor series for arcsin
∞
X
−1/2 z 2k+1
(29)
arcsin z =
(−1)k
,
|z| ≤ 1, z 6= ±1.
k
2k + 1
k=0
we derive by integrating
0
2 −1/2
(arcsin z) = (1 − z )
∞ X
−1/2
=
(−z 2 )k .
k
k=0
If z = ±1, then the series (29) converges, but the points are outside the domain of definition of complex
arcsin.
7.10. Remark. Let f be a holomorphic function in an open set G containing 0 and R be the radius of
convergence of its Taylor series centered at the origin. Then f has a holomorphic continuation at least
up to U (0, R) and there exists a point w with |w| = R such that f has no holomorphic continuation
across w.
However, generally, there is no relation between the convergence of the Taylor series of f at a point z
with |z| = R and the possibility of holomorphic continuation of f across the point z.
8. Laurent series and residues
8.1. Laurent series. There is a possibility of expansion of a holomorphic function on an annulus
{z ∈ C : r < |z − w| < R},
0≤r<R≤∞
into the so called Laurent series, whose terms are an (z − w)n with n ∈ Z (now n can be negative). For
simplicity, we will consider only the punctured disc
P (w, r) = {z ∈ C : 0 < |z − w| < r}.
8.2. Theorem (on Laurent expansion). Let f be holomorphic in P (w, r). Then there exist an ∈ C,
n ∈ Z, such that
∞
X
f (z) =
an (z − w)n ,
z ∈ P (w, r).
n=−∞
11
8.3. Residue. Let f be holomorphic in P (w, r). Then Residue of f at w is defined as the coefficient a−1
of the Laurent expansion
∞
X
f (z) =
an (z − w)n ,
z ∈ P (w, r),
n=−∞
We denote it by res(f ; w).
8.4. Theorem (Residue theorem). Let G ⊂ C be an open set, ϕ be a cycle, hϕi ∪ Int ϕ ⊂ G. Let
M ⊂ Int ϕ be a finite set and indϕ w = 1 for all w ∈ M . Let f be a function holomorphic in G \ M .
Then
Z
X
f (z) dz = 2πi
res(f ; w).
ϕ
w∈M
9. Complex analysis methods of interal calculus
9.1. Example.
Z
∞
sin x
dx.
x
I=
0
The integral is understood in the sense
R
Z
I = lim
R→∞
Since the function
sin x
x
0
sin x
dx.
x
is even, we have
R
Z
2I = lim
R→∞
−R
sin x
dx.
x
We will integrate
f (z) =
eiz − 1
z
over the path
ϕ := ϕ1 +̇ ϕ2 ,
where
• ϕ1 is the line segment from −R to R,
• ϕ2 is the circular arc centered at the origin oriented counter-clockwise, from R to −R.
If we complete the definition of f by f (0) = 1, the function f is holomorphic in C. Indeed, the Taylor
series of eiz − 1 is clearly divisible by z. By Cauchy theorem
Z
f (z) dz = 0.
ϕ
We have
Z R
cos x − 1
sin x
dx + i
dx → 2iI;
x
ϕ1
−R
−R x
where the arrow means the passage to a limit for R → +∞. The first integral on the right vanishes as
the integrand is odd. Further,
Z
Z
Z
Z
eiz
dz
eiz
f (z) dz =
−
=
− πi → −πi.
ϕ2 z
ϕ2 z
ϕ2
ϕ2 z
Z
Z
R
f (z) dz =
Hence
I=
π
.
2
9.2. Example. We have to compute
Z
+∞
I=
e−
x2
2
cos x dx.
−∞
We shall integrate
f (z) = e−z
2
/2
over the path
ϕ := ϕ1 +̇ ϕ2 +̇ ϕ3 +̇ ϕ4 ,
where
12
•
•
•
•
ϕ1
ϕ2
ϕ3
ϕ4
is
is
is
is
the
the
the
the
line
line
line
line
segment
segment
segment
segment
from
from
from
from
−R to R,
R to R + i,
R + i to −R + i,
−R + i to −R.
By the Cauchy theorem
Z
f (z) dz = 0.
ϕ
We pass to limit for R → +∞. We have
Z
Z ∞
√
x2
e− 2 dx = 2π,
f (z) dz →
−∞
Z ∞
ϕ1
Z
f (z) dz → −
e
1−x2
2
(cos x + i sin x) dx → −e1/2 I.
−∞
ϕ3
Hence
r
I=
2π
.
e
9.3. Example.
+∞
xp−1
dx,
0 < p < 1.
x+1
0
We will use the Cauchy formula at the point 1 for the function
Z
I=
f (z) = z p−1
and the path
ϕ := ϕ1 +̇ ϕ2 +̇ ϕ3 +̇ ϕ4 ,
where
•
•
•
•
ϕ1 is the line segment from re−i(π−ε) to Re−i(π−ε) ,
ϕ2 is the circular arc centered at the origin counter clockwise oriented from Re−i(π−ε) to Rei(π−ε) ,
ϕ3 is the line segment from Rei(π−ε) to rei(π−ε) ,
ϕ4 is the circular arc centered at the origin clockwise oriented from rei(π−ε) to re−i(π−ε) .
Namely,
ϕ1 (t) = te−i(π−ε) ,
t ∈ hr, Ri,
it
t ∈ hε − π, π − εi,
ϕ2 (t) = Re ,
−̇ ϕ3 (t) = tei(π−ε) ,
t ∈ hr, Ri,
it
−̇ϕ4 (t) = re ,
t ∈ hε − π, π − εi.
By the Cauchy formula
Z
2πi = 2πif (1) =
ϕ
z p−1
dz.
z−1
We will pass to the limit first for ε → 0+, then for r → 0 and R → +∞. We obtain
Z
Z R −iπ (p−1) p−1
Z R p−1
Z ∞ p−1
z p−1
e
t
t
t
ε→0+
−iπ
−iπp
−iπp
dz →
e
dt = −e
dt → −e
dt = −e−iπp I,
z
−
1
−t
−
1
t
+
1
t
+1
ϕ1
r
r
0
similarly
Z
ϕ3
z p−1
dz = −
z−1
Z
−̇ϕ3
z p−1
ε→0+
dz → −
z−1
Z
R
r
eiπ (p−1) tp−1 iπ
e dt → eiπp
−t − 1
while
Z
ϕ2
z p−1
dz → 0,
z−1
We infer that
I=
Z
ϕ4
π
.
sin πp
13
z p−1
dz → 0.
z−1
Z
0
∞
tp−1
dt = eiπp I,
t+1
9.4. Example. We compute
Z
+∞
I=
−∞
cos x
dx
x2 + 1
We will integrate
Z
ϕ
eiz
dz,
z2 + 1
over the path
ϕ := ϕ1 +̇ ϕ2 ,
where
• ϕ1 is the line segment from −R to R,
• ϕ2 is the circular arc centered at the origin counter clockwise oriented from R to −R.
We shall assume R > 1. We have two possibilities:
(a) By the Cauchy formula
Z
Z
e−1
g(z)
eiz
π
= 2πi
= 2πig(i) =
dz =
dz,
2
e
i+i
ϕ z−i
ϕ z +1
where
eiz
.
z+i
g(z) =
(b) Using the residue theorem
π
e−1
= 2πi
= 2πi resi f =
e
2i
where
f (z) =
Z
ϕ
eiz
dz,
+1
z2
eiz
.
+1
z2
We have
eiz
dz → I,
+1
ϕ1
as the imaginary part of the integrand is odd. Further
Z
eiz
dz → 0.
2
ϕ2 z + 1
Z
z2
Hence
I=
π
.
e
9.5. Example.
Z
+∞
cos x
dx.
2 + 1)2
(x
−∞
(a) According to the Cauchy formula for the first derivative (Theorem 6.4), using the same path ϕ as
in Example 9.4,
Z
Z
g(z)
eiz
eiz
2πig 0 (i) =
dz
=
dz,
g(z)
=
,
2
2
2
(z + i)2
ϕ (z − i)
ϕ (z + 1)
I=
Letting R → ∞ we obtain
I = 2πig 0 (i).
Since
ieiz
eiz
−2
,
2
(z + i)
(z + i)3
g 0 (z) =
we obtain
I = 2πi
ie−1
(2i)2
−2
e−1 2πi 1
2 π
=−
+
= .
3
(2i)
e 4i 8i
e
(b) Similarly as in Example 9.4, we obtain
I = 2πi resi f,
14
where
eiz
.
(z 2 + 1)2
It remains to evaluate the residue of f at i. First, we will decompose the rational function into partial
fractions. We obtain
2i
1
1
1
1
= 2i
=
−
,
z2 + 1
z−i z+i
z−i z+i
Passing to the square we obtain
1
1 2 1 2 1 2
−4
1 2 1 2
1
1
i
i
=
−
=
+
2
+
=
+
+
−
.
2
z +1
z−i z+i
z−i
z−i z+i
z+i
z−i
z+i
z+i z−i
The residue of the sum we will compute as the sum of residues. The summands where in the denominator
there occurs z + i are holomorphic and thus not relevant. From the Taylor expansion of ei(z−i) centered
at i, namely,
(z − i)2
− ...,
ei(z−i) = 1 + i(z − i) −
2!
we obtain
e−1 ei(z−i)
e−1
e−1 i (z − i)
eiz
=
=
+
+ ...,
(z − i)2
(z − i)2
(z − i)2
(z − i)2
i eiz
i e−1 ei(z−i)
i e−1
=
=
+ ...,
z−i
z−i
z−i
so that
−4 resi f = 2ie−1
and
π
I= .
e
f (z) =
15