Differential equation

1
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• ???? ???

Solution by series (skip) Complex algebra
Lecture 5
2
Infinite series

• 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.

3
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
  is convergent, it is absolutely
  convergent.

4
Power Series

• A power series about x0 is
• 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
• Maclaurin series (about zero)
• Frobenius series

5
Important topics in series

• Method of Frobenius
• The differential equation
• Bessels equation
• The equation arises so frequently in practical
  problems that the series solutions have been
  standardized and tabulated.

can be solved by putting
6
Special Functions

• Bessels equation of order ?
• occurs in studies of radiation of energy and in
  other contexts, particularly those in cylindrical
  coordinates
• Solutions of Bessels equation
• when 2? is not an integer
• when 2? is an integer
• when ? n 0.5
• when ? n 0.5

7
Complex algebra
r
y
?
x
Properties
De Moivres theorem For all rational values of
n,
Note ? is not included!
8
Complex numbers and Trigonometric-exponential
identities
Hyperbolic functions
9
Derivatives of a complex variable
Consider the complex variable
to be a continuous function, and let
and . Then the
partial derivative of w w.r.t. x, is
or
Similarily, the partial derivative of w w.r.t.
y, is
or
Cauchy-Riemann conditions
They must be satisfied for the derivative of a
complex number to have any meaning.
10
Analytic functions

• A function of the complex variable
  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.

11
If w z3, show that the function satisfies the
Cauchy-Riemann conditions and state the region
wherein the function is analytic.
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 ?.)
12
If w z-1, show that the function satisfies the
Cauchy-Riemann conditions and state the region
wherein the function is analytic.
Satisfy! Except from the origin
?
Cauchy-Riemann conditions
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.
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At the origin, y 0
As x tends to zero through either positive or
negative values, it tends to negative infinity.
At the origin, x 0
As y tends to zero through either positive or
negative values, it tends to positive infinity.
Consider half of the Cauchy-Riemann condition
, which is not satisfied at the
origin. Although the other half of the
condition is satisfied, i.e.
14
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.

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Types of singularities

• Three types of singularities exist
• Poles or unessential singularities
• single-valued functions
• Essential singularities
• single-valued functions
• Branch points
• multivalued functions

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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.

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The order of a pole
If w f(z) becomes infinite at the point z a,
we define
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.
??????
?? ???? pole, (a0)
?
? 0 ? a ????pole,? w ? 0 ?? pole ? order ? 3 ? a
?? pole ? order ? 4
Order 1
n ????? 1,?? w ???? pole ???
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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.

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Essential singularities can be distinguished from
poles by the fact that they cannot be removed by
multiplying by a factor of finite value.
Example
infinite at the origin
We try to remove the singularity of the function
at the origin by multiplying zp
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
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.
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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
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.
21
and
A given range, where the function is single
valued the branch
when 0 ? ? ? 2?
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
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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.

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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 -? lt ? lt ?.

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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.
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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

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.
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When w and z are both real (i.e. v y 0)
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.
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Cauchys theorem

• If any function is analytic within and upon a
  closed contour, the integral taken around the
  contour is zero.

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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
Since the curve is closed, each integral on the
right-hand side can be restated as a surface
integral using Stokes theorem
Stokes theorem
But for an analytic function, each integral on
the right-hand side is zero according to the
Cauchy-Riemann conditions
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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
30
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
B
A
D
C
Consider the integration along the curve ACDB
Independent of path
Along AC, x 0, z iy
Along CDB, r 10, z 10ei?
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Evaluate around a circle with its
centre at the origin
Let z rei?
Although the function is not analytic at the
origin,
Evaluate around a circle with its
centre at the origin
Let z rei?
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Cauchys integral formula
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)
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 (Cauchys theorem). The straight
dotted lines joining the outside contour C and
the inner circle ? are drawn very close together
and their paths are synonymous.
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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
0, where ? is small
Cauchys integral formula It permits the
evaluation of a function at any point within a
closed contour when the value of the function on
the contour is known.
34
The theory of residues

• The theory of residues is an extension of
  Cauchys theorem for the case when f(z) has a
  singularity at some point within the contour C.

35
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
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
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 Cauchys theorem.
the residue of the function at the pole
If the pole is not at the origin but at z0
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Evaluate around a circle
centred at the origin
If z lt a, the function is analytic within the
contour
Cauchys theorem
If z gt 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.
non-zero term, residue
37
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
Laurent expansion
multiply both sides by (z-a)
a?z
If a simple pole exits at z a, then g(z)
(z-a)G(z)
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Evaluate the residues of
Two poles at z 3 and z - 4
The residue at z 3 B1 3/(34) 3/7
The residue at z - 4 B1 - 4/(- 4 - 3) 4/7
Evaluate the residues of
Two poles at z iw and z - iw
The residue at z iw B1 eiw/2iw
The residue at z - iw B1 -eiw/2iw
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If the denominator cannot be factorized, the
residue of f(z) at z a is
indeterminate
LHôpitals rule
Evaluate
around a circle with centre at the
origin and radius z lt ?/n
40
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
where n is a finite integer, and F(z) is analytic
at z a.
F(z) can be expanded by the Taylor series
Dividing throughout by (z-a)n
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
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Evaluate around a circle
of radius z gt a.
has a pole of order 3 at z a, and the residue
is