投影片 1 - National Cheng Kung University
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Transcript 投影片 1 - National Cheng Kung University
11. Complex Variable Theory
1.
Complex Variables & Functions
2.
Cauchy Reimann Conditions
3.
Cauchy’s Integral Theorem
4.
Cauchy’s Integral Formula
5.
Laurent Expansion
6.
Singularities
7.
Calculus of Residues
8.
Evaluation of Definite Inregrals
9.
Evaluation of Sums
10. Miscellaneous Topics
Applications
1. Solutions to 2-D Laplace equation by means of conformal mapping.
2. Quantum mechanics.
3. Series expansions with analytic continuation.
4. Transformation between special functions, e.g., H (i x) c K (x) .
5. Contour integrals :
a)
b)
c)
d)
e)
f)
Evaluate definite integrals & series.
Invert power series.
Form infinite products.
Asymptotic solutions.
Stability of oscillations.
Invert integral transforms.
6. Generalization of real quantities to describe dissipation, e.g.,
Refraction index: n n + i k,
Energy: E E + i
1.
Complex Variables & Functions
From § 1.8 :
Complex numbers :
Complex conjugate :
Polar representation :
£ z x i y ; x, y ¡
(Ordered pair of
real numbers )
z* x i y
z r e i r e i 2 n
r
x2 y2
tan 1
e i cos i sin
y
x
modulus
argument
Multi-valued function single-valued in each branch
E.g.,
z1/m r1/m ei 2 n /m
has m branches.
ln z ln r i 2 n
has an infinite number of branches.
2.
z x iy
Cauchy Reimann Conditions
d f z
f z
dz
Derivative :
f z
z 0
z
lim
lim
z 0
f z z f z
z
where limit is independent of path of z 0.
Let
z x
z y
f exists
f z u z i v z
z x i y
f u i v
f u i v
z xi y
v
f
v u
u
i
lim
i
z 0 z
x 0 x
x
x x
u v
u v
f
i
lim
lim i
z 0 z
y 0
y y
y y
lim
u v
x y
&
u
v
y
x
Cauchy- Reimann
Conditions
f z u z i v z
z x iy
f exists
f z f x, y
u v
x y
If the CRCs are satisfied,
i.e.,
f
u
v
y
x
Cauchy- Reimann
Conditions
f
f
x
y u i v x u i v y
x
y
x
y
x
y
u
v
f i x i y
x
x
f u
v
i
z x
x
f exists
&
is independent of path of z 0.
CRCs satisfied.
Analytic Functions
f (z) is analytic in R C
f exists & single-valued in R.
Note: Multi-valued functions can be analytic within each branch.
f (z) is an entire function if it is analytic z C \ {}.
z0 is a singular point of f (z) if f (z) doesn’t exist at z = z0 .
Example 11.2.1.
z2 is Analytic
z x iy
f z z 2 x2 y 2 2i x y u i v
u x2 y 2
v 2x y
v
u
2x
y
x
f exists & single-valued finite z.
i.e., z2 is an entire function.
u
v
2 y
y
x
Example 11.2.2.
z* is Not Analytic
z x iy
f z z* x i y u i v
ux
v y
u
v
1 1
x
y
v
u
0
x
y
f doesn’t exist z, even though it is continuous every where.
i.e., z2 is nowhere analytic.
Harmonic Functions
u v
x y
CRCs
u
v
y
x
By definition, derivatives of a real function f depend only on the local behavior of f.
But derivatives of a complex function f depend on the global behavior of f.
Let
z u iv
is analytic
u v
x y
u
v
y
x
2 u
2 v
2 u
2 v
2
x
x y
y2
yx
2 v
2 u
2 u
2
y
yx
x y
2 v
x2
2 u 2 u
0
2
2
x y
2 v 2 v
0
2
2
x y
i.e., The real & imaginary parts of must each satisfy a 2-D Laplace equation.
( u & v are harmonic functions )
CRCs
u x, y c
Contours of u & v are given by
du
u
u
dx
dy0
x
y
u v
x y
u
v
y
x
v x, y c
dv
v
v
dx
dy0
x
y
Thus, the slopes at each point of these contours are
u
dy
x
mu
u
d
x
u
y
CRCs
mumv 1
v
dy
x
mv
v
d
x
v
y
at the intersections of these 2 sets of contours
i.e., these 2 sets of contours are orthogonal to each other.
( u & v are complementary )
z x iy
Derivatives of Analytic Functions
Let f (z) be analytic around z, then
d f x
g x
dx
d f z
g z
dz
f z
f x i y d f x
x
dx
Proof :
f (z) analytic
d xn
n x n 1
dx
E.g.
g z
xz
d zn
n z n 1
dz
Analytic functions can be defined by Taylor series of the
same coefficients as their real counterparts.
Example 11.2.3.
Derivative of Logarithm
CRCs
u v
x y
r
d ln z 1
dz
z
Proof :
u
v
y
x
y
tan 1
x
x2 y2
for z within each branch.
ln z ln r i 2 n
u iv
u ln r
v 2 n
u 1r
x v
2
x rx r
y
x
y2 1
2
1 2
r
y
x x
u 1 r
y
v
2
y ry r
x
y
y2 y
1 2 2
r2
x
x x
1
1
ln z is analytic within each branch.
d ln z ln z u
v
1
1
x
y
i
2 i 2
dz
x
x
x
xiy z
r
r
r2 z z *
QED
Point at Infinity
Mathematica
The entire z-plane can be mapped 1-1 onto the surface of the unit sphere,
the north (upper) pole of which then represents all points at infinity.
Cauchy’s Integral Theorem
3.
C : z t x t i y t
Contour = curve in z-plane
f z u z i v z
Contour integrals :
t1
dz f z
C
d t x t i y t u t i v t
t0
t1
d t xu yv i xv yu
The t -integrals are just
Reimann integrals
t0
A contour is closed if
Closed contour integral :
z t0 z t1
d z f z
C
( positive sense = counter-clockwise )
Statement of Theorem
A region is simply connected if every closed curve in it
can be shrunk continuously to a point.
Let C be a closed contour inside a simply connected region R C.
If f (z) is analytic in R, then
d z f z 0
C
Cauchy’s intgeral theorem
zn on Circular Contour
Example 11.3.1.
z r ei
C
d z i r e i d
on a circular contour
2
n
n 1
dz z i r
d ei n1
0
r n 1 i n 1
n 1 e
2
i d
0
2
0
n integers & n 1
0
2 i
n 1
n integers & n 1
n 1
zn on Square Contour
Example 11.3.2.
Contour integral
from z = z0 to z = z1 along a straight line:
z t z0 z1 z0 t
t : 0 1
d z z1 z0 dt
z1
1
d z f z d t z z f z z z t
1
z0
0
0
1
0
0
Mathematica
For n = 1, each line segment integrates to i /2.
For other integer n, the segments cancel out in pairs.
C
n 1
2 i
n
z
dz
0 n integers & n 1
d z f z 0
Proof of Theorem
C
z x iy
d z f z u d x v d y i v d x u d y
f ui v
Stokes
theorem :
C
C
d r V dσ V
S
Vx d x Vy d y
S
C
S simply-connected
S
For S in x-y plane :
C
d z f z
S
0
S
V V
dx dy y x
x y
v u
d x d y
i
x y
QED
Note: The above (Cauchy’s) proof
requires xu, etc, be continuous.
Goursat’s proof doesn’t.
S
u v
dx dy
x y
f analytic in S CRCs
u v
x y
u
v
y
x
Multiply Connected Regions
y
R
C
d z f z 0
C
C
C
C
C
C
R
C
C
C
d z f z 0
d z f z d z f z
C
x
Value of integral is unchanged for any continuous
deformation of C inside a region in which f is analytic.
C
Cauchy’s Integral Formula
4.
Let f be analytic in R & C R.
f z0
1
2 i
C
y
f z
dz
z z0
z0 inside region bounded by C.
Cauchy’s Integral Formula
C
R
f z
dz
z z0
C
lim
r 0
C
z z0 r e i
On C :
z0
f z
dz
z z0
C
f z
dz
z z0
d z i r e i d
2
d i r ei
0
2 i f z0
x
f z0
r ei
QED
Example 11.4.1.
I
dz
C
An Integral
1
z z 2
1
f z0
2 i
C
f z
dz
z z0
C = CCW over unit circle centered at origin.
y
(z+2)1 is analytic inside C.
I 2 i
1
z2
i
z0
z = 2
Alternatively
1
11
1
z z 2 2 z z 2
1
1
I
dz
2
z
C
C
1
1
dz
2 i 0 i
z 2
2
x
Example 11.4.2.
Integral with 2 Singular Factors
1
f z0
2 i
C
f z
dz
z z0
y
I
dz
1
4z 1
2
C
C = CCW over unit circle
centered at origin.
1
1
4 z 2 1 2 z 1 2 z 1
1 1
1 1 1
1
2 2z 1 2z 1 4 z 1 / 2 z 1 / 2
I 2 i
1
1 1 0
4
z = 1/2
x
z = 1/2
f z0
Derivatives
f z0
1
2 i
C
f z0
2
2 i
f
n!
z0
2 i
1
2 i
f z
d z z z
C
2
0
3
0
f z
d z z z
C
0
f z
dz
z z0
f analytic in R C.
f z
d z z z
C
f z
dz
z0 z z0
C
n
1
2 i
n 1
f (n) analytic inside C.
f n z0
Example 11.4.3.
Use of Derivative Formula
I
d z z a
sin 2 z
n!
2 i
f z
d z z z
C
0
f analytic in R C.
C = CCW over circle centered at a.
4
C
Let
f z sin2 z
f
3
3!
a
2 i
d z z a
sin 2 z
4
C
f z 2sin z cos z
f z 2 cos 2 z sin 2 z
f 3 z 8sin z cos z
I
2 i 3
8 i
f a
sin a cos a
3!
3
n 1
Morera’s Theorem
Morera’s theorem :
If f (z) is continuous in a simply connected R &
d z f z 0
closed C R,
C
then f (z) is analytic throughout R.
Proof :
z2
d z f z 0
closed C
F
F z2 F z1
d z f z
z1
C
F z2 F z1
1
f z1
z2 z1
z2 z1
1
z2 z1
z2
d z f z f z
1
z1
z2
d z z z1 f z1
z1
F z2 F z1
F z1
z2 z1
z2 z1
f z1 lim
1
z2 z1 f z1
2
i.e., F is analytic in R.
Caution: this fails if R is multiply-connected (F multi-valued).
So is F .
QED
f
Further Applications
f z an z n
If
n
n!
z0
2 i
f z
d z z z
0
C
is analytic & bounded,
n
i.e.,
f z M
an rn M
then
Proof :
on a circle of radius r centered at the origin,
an
1 n
1
f 0
n!
2 i
Cauchy inequality
f z
d z n 1
z
C = circle of radius r .
C
Let
M r max f z
z r
Corollary ( r ) :
an
1
2
dz
z r
f z
z
n 1
1
2
M r
2
r
r n 1
QED
If f is analytic & bounded in entire z-plane, then f = const. ( Liouville’s theorem )
n 1
If f is analytic & bounded in entire z-plane, then f = const. ( Liouville’s theorem )
If f is analytic & non-constant, then at least one singularity in the z-plane.
E.g.,
f (z) = z is analytic in the finite z-plane, but has singularity at infinity.
So is any entire function.
Fundamental theorem of algebra :
n
Any polynomial
P z ak z k
with n > 0 & a 0 has n roots.
k 0
Proof :
If P has no root, then 1/P is analytic & bounded z.
P = const.
P has at least 1 root, say at z = 1 .
( contradiction )
Repeat argument to the n 1 polynomial P / ( z 1 ) gives the next root z = 2 .
This can be repeated until P is reduced to a const, thus giving n roots.
5.
f z
f n z0
Laurent Expansion
1
2 i
d z
n!
2 i
f z
d z z z
0
C
1
f z0
2 i
f z
z z
C
n 1
f z
dz
z z0
C
1
1
1
z z0
1
z z z z0 z z0
z z0
z z0
1
f z
2 i
f z
n0
z z
0
n0
n
d z
C
z z0 f n z
0
n!
1
1
z z0
f z
z z
n0
z z0
z
z
0
n
Mathematica
n 1
n
Taylor series
( f analytic in R C )
Let z1 be the closest singularity from z0 , then the radius of convergence is | z1 z0 |.
i.e., series converges for
z z0 z1 z0
Laurent Series
f n z0
n!
2 i
r z z0 R
C1
C2
C1 :
C2 :
1
1
z z
z z0
n0
1
f z0
2 i
d z f z
z z
1
1
1
z z z z0 z z 0 z z 0
z z0
z z0
n0
d z z z
C
n 1
0
C
Let f be analytic within an annular region
1
f z
2 i
f z
f z
dz
z z0
Mathematica
z z0
z z0
n
n
1
f z
2 i
n0
z z0
n
C1
f z
1
dz
n 1
2 i
z z0
1
z z
n0
0
n 1
C2
n
z
z
dz
0 f z
1
f z
2 i
n0
z z0
C1
1
z z
n0
n
0
n 1
f z
d z z z0 f z
n
C2
f z
n
1
an
2 i
C
1
dz
n 1
2 i
z z0
an z z0
d z
n
n0
z z0
n
n 1
0
n
n
z
z
dz
0 f z
C2
d z
C2
f z
z z0
n 1
Laurent series
f z
z z0
z z
1
1
n 1
C within f ’s region of analyticity
f z
Example 11.5.1.
Laurent Expansion
n
an
1
2 i
C
1
f z
z z 1
z0 0
Consider expansion about
f is analytic for
an z z0
d z
f z
z z0
0 z 1
Expansion via binomial theorem :
1 n
1
1
f z
z z
z 1 z
n0
Laurent series :
1
an
2 i
dz z
C
1
n 1
1
z z 1
2 i
f z zn
n 1
k 0
C
n
n 1
zk
1
d z n 2
z
0 otherwise
n 1
6.
Singularities
Poles :
Point z0 is an isolated singular point if f (z) is analytic
in a neighborhood of z0 except for the point z0 .
Laurent series about z0 exists.
If the lowest power of z z0 in the series is n,
then z0 is called a pole of order n.
Pole of order 1 is called a simple pole.
Pole of order infinity is called an essential singularity.
Essential Singularities
e1/z is analytic except for z = 0.
1/ z
e
0
1 n
1 n
z
z
n n !
n 0 n!
z = 0 is an essential singularity
sin z
sin z is analytic in the finite z-plane .
n0
lim sin z lim
z
t0
1
z 2 n 1
2n 1 !
n
n
2 n 1! t
n
2 n 1
t = 0 or z =
is an essential singularity
A function that is analytic in the finite z-plane except for poles is meromorphic.
E.g., ratio of 2 polynomials, tan z, cot z, ...
A function that is analytic in the finite z-plane is an entire function.
E.g., ez , sin z, cos z, ...
Example 11.6.1.
Consider
Value of z1/2 on a Closed Loop
f z z1/2
Mathematica
around the unit circle centered at z = 0.
z ei
f z ei /2
Starting at A = 0, we have
Branch cut
(+x)-axis.
2 values at each point : f is double-valued
Value of f jumps when branch cut is crossed.
Value of f jumps going around loop once.
Example 11.6.2.
Consider
Another Closed Loop
f z z1/2
around the unit circle centered at z = 2.
z 2 ei
f z r ei / 2
r 10 6cos
tan
2
sin
2 cos
No branch cut is crossed going around loop.
No discontinuity in value of f .
If branch cut is taken as (+x)-axis, f jumps
twice going around loop & returns to the
same value.
Branch cut
(x)-axis.
Mathematica
For
Branch Point
f z z1/2 ,
1. Going around once any loop with z = 0 inside it results in a different f value.
2. Going around once any loop with z = 0 outside it results in the same f value.
Any branch cut must start at z = 0.
z = 0 is called the branch point of f.
Branch point is a
singularity (no f )
The number of distinct branches is called the order of the branch point.
The default branch is called the principal branch of f.
Values of f in the principal branch are called its principal values.
By convention :
f (x) is real in the principal branch.
Common choices of the principal branch are
0 2
&
A branch cut joins a branch point to another singularity, e.g., .
Example 11.6.3.
ln z has an Infinite Number of Branches
z r ei r ei 2 n
n = 0, 1, 2, ...
ln z ln r i 2 n
Infinite number of branches
d ln z 1
dz
z
z = 0 is the branch point (of order ).
Similarly for the inverse trigonometric functions.
z p exp p ln z exp p ln r i exp i 2 pn
p = integers
exp i 2 pn 1
p = rational = k / m
p = irrational
z p is single-valued.
z p is m-valued.
z p is -valued.
Example 11.6.4.
f z z 2 1
1/2
lim z 1
2
1/2
z
z 1
1/2
z 1
1/2
1
lim 2 1
t 0 t
Multiple Branch Points
1/2
2 branch points at z = 1.
1
1 1/2
2 1/2
2 k
lim 1 t
lim Ck t
t0 t
t 0 t
k 0
1 1 1
lim t t 3
t 0 t
2 8
1 simple pole at z = .
Mathematica
Let
z 1 s ei
z 1 t ei
f z st ei / 2
Let the branch cuts for both ( z 1 )1/2
be along the (x )-axis, i.e.,
,
in the principal branch.
Analytic Continuation
f (z) is analytic in R C
f has unique Taylor expansion at any z0 R .
Radius of convergence is distance from z0 to nearest singularity z1 .
1.
Coefficients of Taylor expansion f (n) (z0) .
2.
f (n) (z0) are independent of direction.
f (z) known on any curve segment through z0
is enough to determine f (n) (z0) n.
Let f (z) & g (z) be analytic in regions R & S, respectively.
If f (z) = g (z) on any finite curve segment in R
S,
then f & g represent the same analytic function in R
S.
f ( or g ) is called the analytic continuation of g ( f ) into R (S).
Path encircling both branch points:
f (z) single-valued.
( effectively, no branch line crossed )
Path in between BPs:
f (z) has 2 branches.
( effective branch line = line joining BPs. )
z 1
1/ 2
z 1
1/ 2
Path encircling z = 1 :
f (z) double-valued.
( z = 1 is indeed a branch point )
z
2
1
1/2
Hatched curves
= 2nd branch
curve = path
Path encircling z = +1 :
f (z) double-valued.
( z = +1 is indeed a branch point )
Example 11.6.5.
f1 z z 1
Consider
n
Analytic Continuation
n
z 1 1
n
z i 1
n0
f 2 z i n1 z i
n0
For any point P on the line sement,
P r 1 i
0 r 1
f1 P r 1 ri
n
n
n0
f2 P i
n0
n 1
r r 1 i
n
f1 P r 1 ri
P r 1 i
n
n
n0
f1 P
f 2 P i n1 r r 1 i
n0
1
1
1
1 r 1 ri r 1 i P
f2 P i
1
n0
i 1
1
1
i
ir r 1
1 ir r 1
1
r 1 i
n
1
f1 P
r i 1
f1 & f2 are expansions of the same function 1/z.
Analytic continuation can be carried out for functions
expressed in forms other than series expansions.
E.g., Integral representations.
n