Transcript 化工應用數學 授課教師： 郭修伯 Lecture 5 Solution by series (skip) Complex algebra Infinite series Sn  u1  u2  u3  ... un • They can.

化工應用數學
授課教師： 郭修伯
Lecture 5
Solution by series (skip)
Complex algebra
Infinite series
Sn  u1  u2  u3  ... un
• They can be accepted as solutions if they are
convergent.
– As n, SnS (some finite number), the series
is “convergent”.
– As n, Sn ±, the series is “divergent”.
– In other cases, the series is “oscillatory”.
Properties of infinite series
• If a series contains only positive real numbers or
zero, it must be either convergent or divergent.
• If a series is convergent, then un 0, as n .
• If a series is absolutely convergent, then it is also
convergent.
– If the series | u1 |  | u2 |  | u3 | ... | un | is convergent,
it is absolutely convergent.
Power Series
• A power series about x0 is:

n
a
(
x

x
)
 n
0
n 0
• Not every differential equation can be solved using power
series method. This method is valid if the coefficient
functions in the differential equation are analytical at a point
• Taylor series:
1
ak 
k!
f
(k )
( x0 )
– Maclaurin series (about zero)
• Frobenius series:
y 

nr
c
(
x

x
)
 n
0
n 0
Important topics in “series”
• Method of Frobenius
– The differential equation
2
d
y
dy
x 2 2  xF ( x)  G( x) y  0
dx
dx

can be solved by putting
y   an x n  r
n 0
• Bessel’s equation
– The equation arises so frequently in practical
problems that the series solutions have been
standardized and tabulated.
Special Functions
• Bessel’s equation of order 
x 2 y  xy  ( x 2  2 ) y  0
– occurs in studies of radiation of energy and in other contexts,
particularly those in cylindrical coordinates
– Solutions of Bessel’s equation
• when 2 is not an integer
y( x)  c1 J ( x)  c2 J  ( x)
(1) n
J ( x)   2 n 
x 2 n
n!(n   1)
n 0 2

• when 2 is an integer
– when  = n + 0.5 y( x)  c1 J n0.5 ( x)  c2 J n0.5 ( x)
– when  = n + 0.5
Complex algebra
z  x  yi  r (cos  i sin  )
y
Properties :
r

x
w z  w  z
p  wz
De Moivre’s theorem:
p  wz
For all rational values of n,
(cos  i sin  ) n  cosn  i sin n
q  w/ z
q w/z
Note:  is not included!
y 2 y3
yn
e  1 y 
  ...  ...
2! 3!
n!
y
x 2 ix 3 x 4
e  1  ix  
  ...
2! 3! 4!
 x2 x4
 

x3 x5
 1    ...  i x    ...
2! 4!
3! 5!

 

 cos x  i sin x
ix
Complex numbers
and
Trigonometric-exponential identities
e xiy  e x (cos y  i sin y)
eix  e ix
cos x 
2
eix  e ix
sin x 
2i
Hyperbolic functions
e x  e x
cosix 
 cosh x
2
e x  ex
sin ix  i
 i sinh x
2
sin ix i sinh x
tanix 

 i tanh x
cosix cosh x
Derivatives of a complex variable
Consider the complex variable w  f (z ) to be a continuous function,
and let w  u  iv and z  x  iy .
Then the partial derivative of w w.r.t. x, is:
w u
v

i
x x
x
or
df u
v

i
dz x
x
w df z df


x dz x dz
Similarily, the partial derivative of w w.r.t. y, is:
w u v

i
y y
y
or
w df z
df

i
y dz y
dz
Cauchy-Riemann conditions
u v

x y
i
df u
v

i
dz y
y
and
u
v

y
x
They must be satisfied for the derivative of a complex number to have any meaning.
Analytic functions
• A function w  f (z ) of the complex variable z  x  iy
is called an analytic or regular function within a
region R, if all points z0 in the region satisfies the
following conditions:
– It is single valued in the region R.
– It has a unique finite value.
– It has a unique finite derivative at z0 which satisfies the
Cauchy-Riemann conditions
• Only analytic functions can be utilised in pure and
applied mathematics.
If w = z3, show that the function satisfies the Cauchy-Riemann conditions and
state the region wherein the function is analytic.
w  z 3  ( x  iy)3  x3  3ix2 y  3xy2  iy3
w  u  iv
u
 3x 2  3 y 2
x
u
 6 xy
y
w  ( x3  3xy2 )  i(3x 2 y  y 3 )
v
 3x 2  3 y 2
y
v
 6 xy
x
Satisfy!
Cauchy-Riemann conditions
Also, for all finite values of z, w is finite.
Hence the function w = z3 is analytic in any region of finite size.
(Note, w is not analytic when z = .)
u v

x y
and
u
v

y
x
If w = z-1, show that the function satisfies the Cauchy-Riemann conditions and
state the region wherein the function is analytic.
1
( x  iy)
( x  iy)
w z 

 2
x  iy ( x  iy)(x  iy) x  y 2
1
w  u  iv
u
y2  x2

x ( x 2  y 2 ) 2
u
2 xy
 2
y ( x  y 2 ) 2
w(
x
y
)

i
(
)
2
2
2
2
x y
x y
v
y2  x2
 2
y ( x  y 2 ) 2
v
2 xy
 2
x
( x  y 2 )2
?
For all finite values of z, except of 0, w is finite.
Hence the function w = z-1 is analytic everywhere in the z plane
with except of the one point z = 0.
Satisfy!
Except from the origin
Cauchy-Riemann conditions
u v

x y
and
u
v

y
x
w  u  iv  (
x
y
)

i
(
)
2
2
2
2
x y
x y
At the origin, y = 0
u
u
1
 2
x
x
1
x
As x tends to zero through either positive or negative values, it tends to negative infinity.
At the origin, x = 0
v
1
y
v 1
 2
y y
As y tends to zero through either positive or negative values, it tends to positive infinity.
Consider half of the Cauchy-Riemann condition
u v

, which is not satisfied at the origin.
x y
Although the other half of the condition is satisfied, i.e.
u
v
 0
y
x
Singularities
• We have seen that the function w = z3 is analytic
everywhere except at z =  whilst the function w =
z-1 is analytic everywhere except at z = 0.
• In fact, NO function except a constant is analytic
throughout the complex plane, and every function
except of a complex variable has one or more
points in the z plane where it ceases to be analytic.
• These points are called “singularities”.
Types of singularities
• Three types of singularities exist:
– Poles or unessential singularities
• “single-valued” functions
– Essential singularities
• “single-valued” functions
– Branch points
• “multivalued” functions
Poles or unessential singularities
• A pole is a point in the complex plane at which the
value of a function becomes infinite.
• For example, w = z-1 is infinite at z = 0, and we
say that the function w = z-1 has a pole at the
origin.
• A pole has an “order”:
– The pole in w = z-1 is first order.
– The pole in w = z-2 is second order.
The order of a pole
If w = f(z) becomes infinite at the point z = a, we define:
g ( z)  ( z  a)n f ( z)
where n is an integer.
If it is possible to find a finite value of n which makes g(z) analytic at z = a,
then, the pole of f(z) has been “removed” in forming g(z).
The order of the pole is defined as the minimum integer value of n for which
g(z) is analytic at z = a.
什麼意思呢？
1
w

比如：
在原點為 pole, (a=0)
z
則
( z)n
1
 g ( z)
z
Order = 1
n 最小需大於 1，使得 w 在原點的 pole 消失。
w
1
z 2.4 ( z  a)3.6
在 0 和 a 各有一個pole，則 w
在 0 這個 pole 的 order 為 3
在 a 這個 pole 的 order 為 4
Essential singularities
• Certain functions of complex variables have an infinite
number of terms which all approach infinity as the
complex variable approaches a specific value. These could
be thought of as poles of infinite order, but as the
singularity cannot be removed by multiplying the function
by a finite factor, they cannot be poles.
• This type of sigularity is called an essential singularity and
is portrayed by functions which can be expanded in a
descending power series of the variable.
• Example: e1/z has an essential sigularity at z = 0.
Essential singularities can be distinguished from poles by the fact that
they cannot be removed by multiplying by a factor of finite value.
Example:
we
1/ 2
1
1
1
 1 
 ... 
 ..
2
n
z 2! z
n! z
infinite at the origin
We try to remove the singularity of the function at the origin by multiplying zp
p 2
p n
z
z
z p w  z p  z p 1 
 ... 
 ..
2!
n!
It consists of a finite number
of positive powers of z,
followed by an infinite number
of negative powers of z.
All terms are positive
As z  0, z p w  
It is impossible to find a finite value of p which will
remove the singularity in e1/z at the origin.
The singularity is “essential”.
Branch points
• The singularities described above arise from the
non-analytic behaviour of single-valued functions.
• However, multi-valued functions frequently arise
in the solution of engineering problems.
• For example:
z
w
w z
1
2
z  re
i
1
2
wr e
1
i
2
For any value of z represented by a point on the circumference of the circle in the z
plane, there will be two corresponding values of w represented by points in the w plane.
w z
1
2
z  re
i
w  u  iv
1
2
wr e
1
i
2
ei  cos  i sin 
1
2
1
1
1
u  r cos  and v  r 2 sin 
2
2
u
1
1

cos 
r 2 r
2
u
1
1

r sin 

2
2
v
1
1

sin 
r 2 r
2
v 1
1

r cos 
 2
2
when 0    2

u 1 v

r r 
and
v
1 u

r
r 
A given range, where the function
is single valued: the “branch”
The particular value of z at which
the function becomes infinite or zero
is called the “branch point”.
The origin is the branch point here.
Cauchy-Riemann conditions in polar coordinates
Branch point
• A function is only multi-valued around closed
contours which enclose the branch point.
• It is only necessary to eliminate such contours and
the function will become single valued.
– The simplest way of doing this is to erect a barrier from
the branch point to infinity and not allow any curve to
cross the barrier.
– The function becomes single valued and
analytic for all permitted curves.
Barrier - branch cut
• The barrier must start from the branch point but it
can go to infinity in any direction in the z plane,
and may be either curved or straight.
• In most normal applications, the barrier is drawn
along the negative real axis.
– The branch is termed the “principle branch”.
– The barrier is termed the “branch cut”.
– For the example given in the previous slide, the region,
the barrier confines the function to the region in which
the argument of z is within the range - <  < .
The successive values of a complex variable z can be represented
by a curve in the complex plane, and the function w = f (z) will have
particular value at each point on this curve.
Integration of functions of
complex variables
• The integral of f(z) with respect to z is the
sum of the product fM(z)z along the curve
in the complex plane:
f

lim

z 0
M
( z )z   f ( z )dz
C
where fM(z) is the mean value of f(z) in the length z of the curve;
and C specifies the curve in the z plane along which the integration
is performed.
w  u  iv  f ( z ) and z  x  iy

C

f ( z ) dz 

C

C
(u  iv)(dx  idy)
(udx  vdy)  i  (vdx  udy)
C
When w and z are both real (i.e. v = y = 0):
 udx
C
This is the form that we have learnt about integration; actually,
this is only a special case of a contour integration along the real axis.
Cauchy’s theorem
• If any function is analytic within and upon a
closed contour, the integral taken around the
contour is zero.

C
f ( z )dz  0
If KLMN represents a closed curve and there are no singularities of f(z)
within or upon the contour, the value of the integral of f(z) around the
contour is:

C
f ( z )dz   (udx  vdy )  i  (vdx  udy )
C
C
Since the curve is closed, each integral on the right-hand side can be
restated as a surface integral using Stokes’ theorem:
 v u 
(
udx

vdy
)


C
A  x  y dxdy
 u v 
C (vdx  udy)  A  x  y dxdy
Stokes’ theorem
 Q P 
(
Pdx

Qdy
)

C
A  x  y dxdy
But for an analytic function, each integral on the right-hand side is
zero according to the Cauchy-Riemann conditions
  f ( z )dz  0
C
Integral of f(z) between two
points
• The value of an integral of f(z) between two
points in the complex plane is independent
of the path of integration, provided that the
function is analytic everywhere within the
region containing all of the paths.
Q
P
Show that the value of  z2 dz between z = 0 and z = 8 + 6i is the same
whether the integration is carried out along the path AB or around the
path ACDB.
The path of AB is given by the equation:
3
3
7  24i 2
y x
 z  ( x  iy ) 2  ( x  ix) 2 
x
4
4
16
B
A
3
dz  dx  idx
4
8 6 i
8
7  24i 4  3i 2
 352 936i
2
  z dz  
x dx 
16
4
3
0
0
D
C
Consider the integration along the curve ACDB
Independent of path
Along AC, x = 0, z = iy

10 i
0
z dz  i 
2
10 i
0
y 2 dy 
1000 i
3
 352  936i
3
Along CDB, r = 10, z = 10ei
8 6 i

10i
tan1
z dz   1
2
 
2
3
4
100e2i 10iei d 
 352 64i
3
dz
Evaluate  2 around a circle with its centre at the origin
C z
Let z = rei
i
i
2
dz
ire d i
i e 
i
C z 2  0 r 2e2i  r 0 e d  r   i   0
0
2
2
dz
Although the function is not analytic at the origin,  2  0
C z
Evaluate
dz
C z around a circle with its centre at the origin
Let z = rei
i
2 ire d
dz
2




i

0  2i
C z 0 rei
Cauchy’s integral formula
C

a
A complex function f(z) is analytic upon and within the solid line
contour C. Let a be a point within the closed contour such that f(z) is
not zero and define a new function g(z):
f ( z)
g ( z) 
za
g(z) is analytic within the contour C except at the point a (simple pole).
If the pole is isolated by drawing a circle  around a and joining  to C,
the integral around this modified contour is 0 (Cauchy’s theorem).
The straight dotted lines joining the outside contour C and the inner
circle  are drawn very close together and their paths are synonymous.
Since integration along them will be in opposite directions and g(z) is
analytic in the region containing them, the net value of the integral
along the straight dotted lines will be zero:
f ( z )dz
f ( z )dz


0
C za
 za
Let the value of f(z) on  be f ( z )  f (a)   ; where  is a small quantity.

C
f ( z )dz
f (a)dz
dz


0
 za
 za
za
0, where  is small
z  a  rei
if (a) 
2
0
re i d
 2if (a)
i
re
1
f ( z )dz Cauchy’s integral formula: It permits the evaluation
 f (a) 
of a function at any point within a closed contour when

C
2i
z  a the value of the function on the contour is known.
The theory of residues
• The theory of residues is an extension of
Cauchy’s theorem for the case when f(z) has
a singularity at some point within the
contour C.
If a coordinate system with its origin at the singularity of f(z) and no
other singularities of f(z). If the singularity at the origin is a pole of
order N, then:
g ( z)  z N f ( z)
will be analytic at all points within the contour C.
g(z) can then be expanded in a power series in z and f(z) will thus be:
BN BN 1
B1 
f ( z )  N  N 1  ...    Cn z n
z
z
z n 0
Laurent expansion of the complex function
The infinite series of positive powers of z is analytic within and upon
C and the integral of these terms will be zero by Cauchy’s theorem.
the residue of the function at the pole

C
If the pole is not at the origin but at z0
f ( z )dz  2iB1
z  ( z  z0 )
e z dz
Evaluate C
3 around a circle centred at the origin
( z  a)
If |z| < |a|, the function is analytic within the contour
e z dz
C ( z  a)3  0
Cauchy’s theorem
If |z| > |a|, there is a pole of order 3 at z = a within the contour.
Therefore transfer the origin to z = a by putting  = z - a.
e z dz
e  a d
1
1 
1 1

a
C ( z  a)3  C 3  e C  3  2  2!  3!  4!  ...d
1 a
non-zero term, residue = e
2

C
1 a
f ( z )dz  2iB1  2i ( e )  ie a
2
Evaluation of residues without
the Laurent Expansion
The complex function f(z) can be expressed in terms of a numerator
and a denominator if it has any singularities:
f ( z) 
F ( z)
g ( z)
f ( z) 
B1
 b0  b1 ( z  a)  ...  bn ( z  a) n  ... Laurent expansion
za
multiply both sides by (z-a)
f ( z)(z  a)  B1  b0 ( z  a)  b1 ( z  a)2  ... bn ( z  a)n1  ...
az
B1  f ( z)(z  a) |z a
If a simple pole exits at z = a, then g(z) = (z-a)G(z)
F (a)
B1 
G (a)
z
Evaluate the residues of z 2  z  12
z
z
f ( z)  2

z  z  12 ( z  4)(z  3)
B1 
F (a)
G (a)
The residue at z = 3:
B1= 3/(3+4) = 3/7
The residue at z = - 4:
B1= - 4/(- 4 - 3) = 4/7
Evaluate the residues of
ez
z 2  w2
ez
ez
f ( z)  2

2
z w
( z  iw)(z  iw)
B1 
Two poles at z = 3 and z = - 4
F (a)
G (a)
The residue at z = iw:
B1= eiw/2iw
The residue at z = - iw:
B1= -eiw/2iw
Two poles at z = iw and z = - iw
If the denominator cannot be factorized, the residue of f(z) at z = a is
( z  a) F ( z )
B1 
g ( z)
z a
0

0
indeterminate
L’Hôpital’s rule
d ( z  a) F ( z ) / dz
B1 
dg( z ) / dz
Evaluate
z a
ez
F ( z)
C sin nzdz  g ( z)
F ( z )  ( z  a) F ' ( z )

g ' ( z)
z a
F ( z)

g ' ( z)
around a circle with centre at the origin
and radius |z| < /n
B1 
ez
d
sin nz
dz
z 0
ez

n cos nz
z 0
1

n
ez
1

dz  2i 
C sin nz
n
Evaluation of residues at multiple poles
If f(z) has a pole of order n at z = a and no other singularity, f(z) is:
f ( z) 
F ( z)
( z  a) n
where n is a finite integer, and F(z) is analytic at z = a.
F(z) can be expanded by the Taylor series:
( z  a) 2
( z  a) n n
F ( z )  F (a)  ( z  a) F ' (a) 
F ' ' (a)  ... 
F (a)  ...
2!
n!
Dividing throughout by (z-a)n
F (a)
F ' (a)
F n1 (a)
f ( z) 

 ... 
 ...
n
n 1
( z  a) ( z  a)
(n  1)!( z  a)
The residue at z = a is the
coefficient of (z-a)-1
The residue at a pole of order n situated at z = a is:
F n1 (a)
B1 
(n  1)!
z a


1 d n1
n

(
z

a
)
f ( z ) z a
n 1
(n  1)! dz
cos 2 z
Evaluate 
dz around a circle of radius |z| > |a|.
C ( z  a)3
cos 2 z
( z  a)3
has a pole of order 3 at z = a, and the residue is:
F n 1 (a)
B1 
(n  1)!
1 d2

2! dz2
z a


1
d n 1
n

(
z

a
)
f ( z ) z a
n 1
(n  1)! dz

3 cos 2 z 
( z  a) ( z  a) 3   2 cos 2a

 z a
cos 2 z

dz  2i(2 cos 2a)
3
C ( z  a)