# 1 Theorems - math.umn.edu

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Oct 8, 2013 (4 years and 9 months ago)

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1 Theorems
Theorem 1 (Arguement Principle) Let D be a open domain and f meromorphic in D with poles
p
1
,p
2
,...p
m
and zeros z
1
,...,z
n
counted according to multiplicity.If γ is a closed rectiﬁable curve in D
with γ ￿= 0 then
1
2πi
￿
γ
f
￿
(z)
f(z)
dz =
n
￿
k=0
n(γ;z
k
) −
m
￿
k=0
n(γ;p
k
)
It is worth noting that this is also gives the winding number of the image of a curve.
Theorem 2 (Binomial Series) Let α be any non zero complex number.Deﬁne the binomial coeﬃcients
as usual
￿
α
n
￿
=
α(α −1) ∙ ∙ ∙ (α −n +1)
n!
Then
(a +b)
α
=

￿
n=0
￿
α
n
￿
a
n
b
α−n
Theorem 3 (Cauchy’s Theorem) 1.If f(z) is analytic in Ω then
￿
γ
f(z)dz = 0
for every cycle γ which is homologous to zero in Ω.
2.Let D be an open disc and let f be analytic on D.Let γ be a closed curve in D.For any point z
0
not on γ
n(γ,z
0
)f(z
0
) =
1
2πi
￿
γ
f(z)
z −z
0
dz
Theorem 4 (Casorati-Weierstrass Theorem) Suppose z
0
is an essential isolated singularity of f(z).
Then for every complex number w
0
,there is a sequence z
n
→z
0
such that f(z
n
) →w
0
Theorem 5 (Conformal Mapping Theorem) If f(z) is analytic at z
0
and f
￿
(z
0
) ￿= 0 then f(z) is
conformal at z
0
.(converse also holds)
Theorem 6 (Cauchy Hadamard forumla) For the series
￿
a
k
z
k
The radius of convergence R is given by
R =
1
limsup
k
￿
|a
k
|
1
Theorem 7 (Cauchy Riemann) Let f = u+iv be deﬁned on a domain D in the complex plane,where
u and v are real-valued.Then f(z) is analytic on D if and only if u(x,y) and v(x,y) have continuous
ﬁrst-order partial derivatives that satisfy the Cauchy Riemann equations
Theorem 8 (Exerior Domain Residue Theorem) Let D be an exterior domain with piecewise smooth
boundary.Suppose that f(z) is analytic on D ∪ ∂D,except for a ﬁnite number of isolated singularities
z
1
,...,z
m
in D and let a
−1
be the coeﬃcent of the 1/z in the laurent expansion of f(z) which converges
for |z| > R.Then
￿
∂D
f(z)dz = −2πa
−1
+2πi
m
￿
j=1
Res[f(z),z
j
]
Theorem 9 (Fractional Residue Theorem) If z
0
is a simple pole of f(z),and C
￿
is an arc of the
circle {|z −z
0
| = ￿} of angle α,then
lim
￿→0
￿
C
￿
f(z)dz = αiRes[f(z),z
0
]
Theorem 10 (Fundamental Theorem of Algebra) If p(z) is a non-constant polynomial then there
is a complex number a with p(z) = 0
Theorem 11 (Green’s Theorem) Let p = p(x,y) and q = q(x,y) be continuous functions on the
closure of a bounded open set U whose boundary consists of a ﬁnite number of continuous curves oriented
so that U lies to the left of each one of these curves.Let C be the boundary of U.Then
￿
C
pdx +qdy =
￿￿
U
￿
∂q
∂x

∂p
∂y
￿
dydx
Theorem 12 (Harnak’s Inequality) for a positive harmonic function u(z)e on a disk |z| < R
ρ −r
ρ +r
u(0) ≤ u(z) ≤
ρ +r
ρ −r
u(0)
Theorem 13 (Harmonic Classiﬁcation Theorem) Any function harmonic in an annulus r < |z| <
R can be written
f(z) = a
0
+b
0
log |z| +

￿
n=−∞
(c
n
z
n
+d
n
¯z
n
)
Theorem 14 (Hurwitz’s Theorem) Suppose {f
k
(z)} is a sequence of analytic functions on a domain
D that converges normally on D to f(z),and suppose that f(z) has a zero of order N at z
0
.Then
there exists ρ > 0 such that for k large,f
k
(z) has exactly N zeros in the disk {|z −z
0
| < ρ} counting
multiplicity,and these zeros converge to z
0
as k →∞.
Theorem 15 (Implicit Function Theorem) Let F(z,w) be a continous function of z and w that
depends analytically on z for each ﬁxed w,and let F
1
(z,w) denote the derivative of F(z,w) with respect
to z.Suppose F(z
0
,w
0
) = 0 and F
0
(z
0
,w
0
) ￿= 0.Choose ρ such that F(z,w
0
) ￿= 0 for 0 < ρ ≤ ρ.Then
2
• there exists δ > 0 such that if |w −w
0
| < δ,there is a unique z = g(w) satisfying |z −z
0
| ≤ ρ and
F(z,w) = 0.
• g(w) =
1
2πi
￿
|ζ−z
0
|=ρ
ζF
1
(ζ,w)
F(ζ,w)
dζ.
• Suppose further that F(z) is analytic in w for each ﬁxed z,and let F
2
(z,w) denote the derivative
of F(z,w) with respect to w.Show that g(w) is analytic and
g
￿
(w) = −F
2
(g(w)),w)/F
1
(g(w),w)
Theorem 16 (Jordan’s Lemma) If Γ
R
is the semicircular contour z(θ) = Re

,for 0 ≤ θ ≤ π,in the
upper half-plane,then
￿
Γ
R
￿
￿
e
iz
￿
￿
|dz| ≤ π
Theorem 17 (Liouville’s Theorem) Let f(z) be an analytic function on the complex plane.If f(z)
is bounded,then f(z) is constant.
Theorem 18 (Maximum Principle) If f(z) is analytic,non-constant in a region Ω,then |f(z)| has
no maximum on Ω.
If f(z) is deﬁned an continuous on a closed bounded set E,and analytic on the interior of E,then
the maximum of |f(z)| on E is assumed on the boundary of E.
Theorem 19 (Mean Value Theorem) Let u be a harmonic function on an open set U.Let z
0
∈ U,
and let r > 0 be a number such that the closed disc of radius r centered at z
0
is contained in U.Then
u(z
0
) =
1

￿
0
u(z
0
+re

)dθ
Theorem 20 (Mittag-Leﬄer Theorem) Let D be a domain in the complex plane.Let {z
k
} be a
sequence of distinct points in D with no accumulation point in D,and let P
k
(z) be a polynomial in
1/(z − z
k
).Then there is a meromorphic function f(z) on D whose poles are the points z
k
such that
f(z) −P
k
(z) is analytic at z
k
.
Theorem 21 (ML-estimate) Suppose γ is a piecewise smooth curve.If h(z) is a continuous function
on γ,then
￿
￿
￿
￿
￿
γ
h(z)dz
￿
￿
￿
￿

￿
γ
|h(z)| |dz|
Further,if γ has length L,and |h(z)| ≤ M on γ then
￿
￿
￿
￿
￿
γ
h(z)dz
￿
￿
￿
￿
≤ ML
3
Theorem 22 (Monodromy Theorem) Let U be a connected open set.Let f be analytic at a point z
0
of U,and let γ and η be two paths from z
0
to w ∈ U.Further assume:
• γ is homotopic to η in U.
• f can be continued analytically along any path in U.
Let f
γ
and f
η
be the analytic continuations of f along γ and η respectively.Then f
γ
and f
η
are equal
in some neighborhood of w.
Theorem 23 (Morera’s Theorem) Let f(z) be a continuous function on a domain D.If
￿
∂R
f(z)dz = 0
For every closed rectangle R contained in D with sides parallel to the coordinate exes,then f(z) is analytic
on D.
Theorem 24 (Picard’s Big Theorem) Suppose f(z) is meromorphic on a punctured neighborhood
{0 < |z −z
0
| < δ} of z
0
.If f(z) omits three values at z
0
then f(z) extends to be meromorphic at z
0
.
Theorem 25 (Picard’s Little Theorem) A non constant entire function assumes every value in the
complex plane,with at most one exception.
Theorem 26 (Pick’s Lemma) If f(z) is analytic and satisﬁes |f(z)| < 1 for |z| < 1 then
|f
￿
(z)| ≤
1 −|f(z)|
2
1 −|z|
2
If f(z) is a conformal self-map of D,then equality holds,otherwise there is strict inequality for all
|z| < 1.
Theorem 27 (Poisson’s Forumla) Let h(e

) be a continuous function on the unit circle.Then the
Poisson integral
˜
h(z) deﬁned by
˜
h(z = re

) =
π
￿
−π
h(e

)P
r
(θ −ϕ)

x is a harmonic function on the open unit disk that has boundary values h(e

).
Theorem 28 (Prime Number Theorem) Let π(x) be the number of primes ≤ x.
π(x) ≈
x
log x
Theorem 29 (Open Mapping Theorem) Let D be a region and suppose that f is a non-constant
analytic function on G.Then for any open set U in G f(U) is open.
4
Theorem 30 (Reﬂection Principle) Let D
+
be the part in the upper half plane of a region D and let
σ be the part of D on the real-axis.Suppose v(x) is continuous in D
+
∪σ,harmonic in D
+
,and 0 on σ.
Then v has a harmonic extension to D which satisﬁes the symmetry relation v(¯z) = −v(z).Likewise if
f(z) is analytic on D
+
then f(z) has a analytic extension which satisﬁes f(z) =
¯
f(¯z)
Theorem 31 (Riemann Mapping Theorem) Given any simply connected region D which is not the
whole plane,and a point z
0
∈ D,there exists a unique analytic function f(z) in D,normalized by the
conditions f(z
0
) = 0,f
￿
(z
0
) > 0,such that f(z) deﬁnes a injective mapping of D to the disk |w| < 1
Theorem 32 (Residue Theorem) Let U be an open set,and γ a closed chain in U such that γ is null-
homologous in U.Let f be analytic on U except at a ﬁnite number of points z
1
,...,z
n
.Let m
i
= W(γ,z
i
).
Then
￿
γ
f = 2πi
n
￿
k=1
m
k
∙ Res
z
k
f
Theorem 33 (Rouche’s Theorem) Let γ be a closed path homologous to 0 in U and assume that γ
has an interior.Let f,g be analytic on U with |f(z) −g(z)| < |f(z)| for z on γ.Then f(z) and g(z)
have the same number of zeros in the interior of γ.
Theorem 34 (Runge’s Theorem) Let K be a compact subset of the complex plane.If f(z) is analytic
on an open set containing K,then f(z) can be approximated uniformly on K be rational functions with
no poles on K.
Theorem 35 (Schwarz Lemma) Let f(z) be analytic for |z| < 1.Suppose |f(z)| ≤ 1 for all |z| ≤ 1
and f(0) = 0.Then
|f(z)| ≤ |z| |z| < 1
Further,if equality holds at some point z
0
￿= 0,then f(z) = λz for some constant λ of unit modulus.
Theorem 36 (Schwarz-Christoﬀel Forumla) The functions z = F(w) which map |w| < 1 confor-
mally onto polygons with angles α
k
π(k = 1,...,n) are of the form
F(w) = C
w
￿
0
n
￿
k=1
(w −w
k
)
−β
k
dw +C
￿
Where β
k
= 1 −α
k
,the w
k
points are on the unit circle,and C,C
￿
are complex constants.
Theorem 37 (Uniqueness Principle) If f(z) and g(z) are analytic on a domain D and if f(z) = g(z)
for all z in a set that has a non-isolated point,then f(z) = g(z) for all z ∈ D.
Theorem 38 (Weierstrass M-test) Let (M
k
) be a sequence of nonnegative real numbers such that
￿
M
k
< ∞.If |g
k
(z)| ≤ M
k
for all x in a set S,then
￿
g
k
converges uniformly on S.
Theorem 39 (Weierstrass Product Theorem) Let D be a domain in the complex plane.let {z
k
}
be a sequence of distinct points of D with no accumulation point in D,and let {m
k
} be a sequence of
integers.Then there is a meromorphic function f(z) on D whose only zeros and poles are at the points
z
k
such that the order of f(z) at z
k
is n
k
.
5
2 Deﬁnitions
Analytic:A function f(z) is analytic on the open set U if f(z) is (complex) diﬀerentiable at each point
of U and the complex derivative f
￿
(z) is continuous on U.
Bernoulli Number:The Bernoulli Numbers are a sequence of signed rational numbers given by the
identiety
x
e
x
−1
=
￿
n = 0

B
n
x
n
n!
Closed:ω ∈
￿
k
M is closed if dω = 0
Conformal Mapping:A function is conformal if it preserves angles.
Cauchy Riemann Equations:Let f = u(x,y) +iv(x,y) be a complex function of a complex variable.
∂u
∂x
=
∂v
∂y
∂u
∂y
= −
∂v
∂x
If f(z) is a complex function of a complex variable,the Cauchy Riemann equation can be better
written as
∂f
∂¯z
= 0
Cross Ratio:The Cross Ratio (z
1
,z
2
,z
3
,z
4
) is the image of z
1
under the linear transformation which
caries z
2
,z
3
,z
4
to 1,0,∞.
(z
1
,z
2
,z
3
,z
4
) =
(z
1
−z
3
)(z
2
−z
4
)
(z
2
−z
3
)(z
1
−z
4
)
Conformally Equivalent:We say that two domains are conformally equivalent if there is a conformal
map of one onto the other.
Elliptic Function:Let L be a lattice.An elliptic function f (with respect to L) is a meromorphic
function on C which is L-periodic.
f(z +ω) = f(z)
for all z ∈ C and ω ∈ L.
Exact:A smooth diﬀerential form ω ∈
￿
k
M is exact if there exists a smooth (k −1)-form η on M such
that dη = ω.
Exterior Domain:A Exterior Domain is a domain D in the complex plane that includes all z such
that |z| ≥ R for some R.
Fractional Linear Transformation:A mapping of the form
S(z) =
az +b
cz +d
Harmonic:Let u(x,y) be a C
2
function.u is harmonic if it satisﬁes the equation:
6

2
u
∂x
2
+

2
u
∂y
2
= 0
Holomorphic:A function f deﬁned on an open set U ⊆ C is said to be diﬀerentiable if it is (complex)
diﬀerentiable at every point z ∈ U.
Hyperbolic Geometry/Metric:
ds =
￿
dx
2
+dy
2
y
Laurent Series:A series
f(z) =

￿
n=−∞
a
n
z
n
M¨obius Transformation:A mapping of the form
S(z) =
az +b
cz +d
Meromorphic:A function f on a domain D is meromorphic if f(z) is analytic on D except at isolated
singularities,each of which is a pole.
Poisson Integral:The Poisson Integral
˜
h(z) of h(e

) is a function on the open unit disk,deﬁned by
˜
h(z) =
π
￿
−π
h(e

)P
r
(θ −ϕ)

Where z = re

.
Poisson Kernel:The Poisson Kernel is deﬁned by:
P
r
(θ) =

￿
k=−∞
r
|r|
e
ikθ
Pole:An isolated singularity of f(z) for which there exists some N > 0 such that a
−N
￿= 0 but a
k
= 0
for k < −N.The integer N is the order of the pole.
Residue:Let f(z) have a Laurent expansion at a point z
0
.
f(z) =

￿
k=−∞
a
k
(z −z
0
)
k
We call a
−1
the residue of f at z
0
.
Riemann Map:Let D be a domain and D be the unit disk.A Riemann Map is a conformal map
ϕ:D →D
singularities:Let f(z) be a function with Laurent Series
f(z) =

￿
k=−∞
a
k
(z −z
0
)
k
7
• Isolated Singularity:A point z
0
is an isolated singularity of f(z) if f(z) is analytic in some
punctured disc {0 < |z −z
0
| < r} centered at z
0
.
• Removable Singularity:The isolated singularity at z
0
is a removable singularity if a
k
= 0 for
k < 0.i.e.has the form:
f(z) =

￿
k=0
a
k
(z −z
0
)
k
• Essential Singularities:A isolated singularity for which a
k
￿= 0 for inﬁnitely many k < 0.(A
singularity which is neither removable,nor a pole.
Winding Number:If γ is a closed path we deﬁne the winding number at a point α to be
W(γ,α) =
1
2πi
￿
γ
1
z −α
dz
3 Special Functions
Bessel Function:
J
n
(z) =
1
2πi
￿
γ
ζ
−n−1
e
z/2(ζ−1/ζ)

Where γ is the unit circle centered at 0 and oriented in the counterclockwise direction.
The function can also be deﬁned as a solution to the diﬀerential equation
w
￿￿
+
1
z
w
￿
+
￿
1 −
n
2
z
2
￿
w = 0
The Bessel function as the power series
J
n
(z) =

￿
k=0
(−1)
k
z
n+2k
k!(n +k)!2
n+2k
Elliptic Modular Function:
λτ =
e
3
−e
2
e
1
−e
2
Where e
1
,e
2
,e
3
are the roots of the equation
4w
3
−g
2
w −g
3
Exponential Function:If z = x +iy
e
z
=

￿
b=0
z
n
n!
= e
x
cos y +ie
x
siny
Gamma Function:
8
Γ(z) =
e
−γz
z

￿
n=1
￿
1 +
z
n
￿
−1
e
z/n
Also often given as
Γ(z) =
1
￿
0
￿
log(
1
t
)
￿
z−1
dt
=

￿
0
t
z−1
e
−t
dt
Logarithm Function:
log z = log |z| +i arg z
Probablity Integral:

￿
0
e
−t
2
dt =

π
2
Product Forms of Trig functions:
sinπz =
￿
n￿=0
￿
1 −
z
2
n
2
￿
Weierstrass ℘ function:
℘(z) =
1
z
2
+
￿
ω￿=0
￿
1
(z −ω)
2

1
ω
2
￿
Where ω = n
1
ω
1
+n
2
ω
2
￿
d℘
dz
￿
2
= 4℘
3
−g
2
℘ −g
3
Weierstrass ζ function:
ζ(z) =
1
z
+
￿
ω￿=0
￿
1
z −w
+
1
w
2
+
z
ω
2
￿
Riemann ζ Function:
ζ(s) =

￿
n=1
1
n
s
9