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Final Answers
© 2000-2014   Gérard P. Michon, Ph.D.

Analysis, Complex Analysis

 

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Complex Variables, Complex Analysis   by  John H. Mathews  (2000).
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Convergence, Series, Complex Analysis


brentw (Brent Watts, NC. 2000-12-07)   Defining  Real  Numbers...
What's the definition of a Cauchy sequence?

A Cauchy sequence is a sequence  U  whose  far terms  are within a vanishing distance of each other...  This is to say that, given any small positive quantity e, there's an integer N(e) such that, for any p and q both larger than N(e), the distance from U(p) to U(q)  (usually:  |U(p)-U(q)| ) is always less than e.

In this, the distance  |x-y|  is usually based on ordinary norms, as you'd expect, but more exotic possibilities exist, including the p-adic metric.

 Augustin Cauchy 
 1789-1857 The concept was first introduced by Augustin Cauchy (1789-1857)  as a nice way to characterize convergent sequences of real numbers  without  referring explicitely to the  limits  they converge to.  ( Cours d'analyse de l'Ecole Polytechnique,  1821.)

A convergent sequence is always a  Cauchy sequence.  The converse is only true in a  complete  space  (like the real numbers); it's not true for the rationals.  In fact a complete space can be defined as a metric space in which every Cauchy sequence converges.

Following Georg Cantor (1845-1918) one usually defines real numbers, as equivalence classes of rational Cauchy sequences.  Two sequences U and V are considered equivalent if the limit of U(n)-V(n) is zero.

For example, the constant sequence  U(n) = 1  is a Cauchy sequence equivalent to the Cauchy sequence  V(n) = 1-(0.1)n ,  whose first terms are:

0     0.9     0.99     0.999     0.9999     0.99999     0.999999     0.9999999

So, both sequences define the  same  real number  (the number 1).  Any real number which has a finite decimal expansion also has an infinite one, ending with infinitely many nines.  Other  numbers have just a  single  decimal expansion.  This confuses many beginners, as they wrestle with the  definition  of real numbers.

There's another (equally valid) way to define real numbers, which predates the above.  It's based on  Dedekind cuts  which turn out to be more difficult to manipulate than Cauchy sequences...

On 2001-03-11, Brent asked for:
[A specific] example of how to prove whether a given sequence is a Cauchy sequence or not.

In the realm of real numbers, proving that a sequence converges and proving it's a Cauchy sequence are just two aspects of the same thing.  Therefore, we'll choose an example of a sequence in the the field of rationals (a notoriously incomplete space, as was first glimpsed by a disciple of Pythagoras, probably Hippasus of Metapontum, about 2500 years ago).

Consider the rational sequence u, recursively defined via:

u(0)  =  1       and       u(n+1)   =   u(n) / 2  +  1 / u(n)
 
u(1) = 3/2 ,  u(2) = 17/12 ,  u(3) = 577/408 ,  u(4) = 665857/470832   etc.

First you may want to prove that u(2n) is an increasing sequence and that u(2n+1) is a decreasing one, whereas u(2m+1) is greater than any u(2n) for any pair n,m. With the additional remark that u(2n+1)-u(2n) tends toward zero as n tends to infinity, you've got all the ingredients to prove that, for p and q greater than n, |u(p)-u(q)|  is less than  |u(n)-u(n+1)|  and thus tends to zero when n tends to infinity.  In other words, the sequence u is a rational Cauchy sequence

On the irrationality of the constant of Pythagoras

The above should come as no surprise to anyone who knows about the irrational limit of u (namely Ö2), a "special" number which was not at all taken for granted 2500 years ago: The irrationality of what is still sometimes referred to as the  constant of Pythagoras  is said to have prompted the sacrifice to the gods of 100 oxen (a so-called hecatomb)...


johnrp (John P. of Middletown, NJ. 2000-10-14)
Can you rearrange the following infinite series so that its sum equals 43?
1 -1/2 +1/3 -1/4 +1/5 -1/6 +1/7 -1/8 +1/9 -1/10 ...

Yes.  In fact such a thing can be done for any "target sum" S (here S=43) with any series which is convergent but not absolutely convergent (that is, the series of absolute values does not converge).  This result is known as the Riemann Series Theorem.

This applies here, because the series involved converges to a well-known constant while the series of absolute values is the  harmonic series, which has been known to diverge since the 14th century (at least).  Let's discuss the construction in general terms:

Take just as many positive terms of the series as necessary to exceed S (that's always possible, as explained below), then take as many negative terms to have the partial sum fall below S, then use positive terms again to go above S, switch to negative terms to go below S again, etc.

Note that, as advertised above, it is always possible to add enough terms of the series to make up for any (positive or negative) difference between the current sum and the target S. That's because the series of the absolute values is divergent (so both the series of negative terms and the series of positive terms must be divergent, or else the whole series would not be convergent).

In this process (at least after the first step) the difference between S and any partial sum never exceeds the magnitude of the term added at the latest "switch" from negative to positive (or vice-versa). Since the magnitudes of such terms tend to zero, partial sums tend toward S. S is therefore the sum of the rebuilt series.

Oresme's proof of the divergence of the harmonic series :

To apply the above to the question at hand, we have to show that the following so-called  harmonic series  diverges:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + ...

   Nicole Oresme 
 1323-1382
One elementary way to do so was published around 1350 by Nicole Oresme (1323-1382).  It's based on the remark that the series is bounded from below by the series obtained by replacing 1/n with 1/p, where p is the lowest power of 2 greater than or equal to n:

1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/16 + 1/16 + ...

By grouping equal terms in this sum, we see that the partial sum up to the term of rank n=2p is simply equal to 1+p/2. The partial sums of the original series up to the same rank are therefore no less than 1+p/2 as well.  This means that such partial sums will eventually exceed any preset goal, no matter how high.  The series (slowly)  diverges ;  its limit is infinityQED

The Harmonic Series Diverges Again and Again   by  Steven J. Kifowit  &  Terra A. Stamps   (2006)
Riemann rearrangement theorem   |   The Mercator series (1668)


(2012-10-25)   Decreasing divergent series whose minimum converges!

Such a counterintuitive monstrosity can be constructed very simply from any convergent series with decreasing positive terms  un

We'll build two divergent series of terms  an  and  bn  such that:

un   =   min ( an , bn )

Let's choose a sequence of indices  p(0) = 0 < p(1) < p(2) < p(3) < ...
such that  p(i+1) - p(i) > 1/ui

  • If   p(2i) ≤ n < p(2i+1) ,  then   an = up(2i)   and   bn = un
  • If   p(2i+1) ≤ n < p(2i+2) ,  then   an = un   and   bn = up(2i+1)

The sum of all terms  an  when  n  goes from  p(2i)  to  p(2i+1)-1  is greater than  1, by construction, so the  an  series diverges!  So does the  bn  series, for similar reasons.  QED

Strictly speaking, the two series described are merely  nonincreasing  but minor details could be adjusted to make them  decreasing  ones.

Two Divergent Series Whose Minimum Converges  (March 2010)


 Nonuniform convergence
 to a function which is 
 zero except when x=1... (Brent Watts of Hickory, NC. 2001-04-13)
How do you show that the sequence fn : x ® xn converges for each x in the closed interval [0,1] but that the convergence isn't uniform?


The simple convergence of a sequence of functions is just pointwise convergence. In this case, the limit of xn is clearly 0 when x is in [0,1[ and 1 when x=1. Your sequence of function fn thus converges and its limit is the function f defined over [0,1] which is zero everywhere except at point 1, where f(1)=1.

Now, simple convergence does not tell you much about the limit. The limit of continuous functions may not be continuous (this is what happens here). Worse, the integral of the limit may not be equal to the limit of the integrals:  A sequence of functions with 
 a unit integral, whose 
 nonuniform limit is zero! Consider, for example, the sequence of functions gn on [0,1] for which gn(x) is n2x when x is in [0,1/n], n(2-nx) when x is in [1/n,2/n] and zero elsewhere. The pointwise limit of gn(x) is always zero (x=0 included, since gn(0)=0 for any n). Yet, the integral of gn is always equal to 1, for any n>1.

This is why the notion of uniform convergence was introduced: We say that a sequence of functions fn defined on some domain of definition D converges uniformly to its limit f when it's always possible for any positive quantity e to exhibit a number N(e) such that whenever n is more than N(e), the quantity |fn(x)-f(x)| is less than e, for any x in D.  (Note that a "domain of definition" is not necessarily a "domain" in the sense of an open region, ita est.  Whenever it's critical, make sure to specify which meaning of "domain" you have in mind.)

Uniform convergence does imply that the integral of the (uniform) limit is the limit of the integrals.  It also implies that the (uniform) limit of continuous functions is continuous. Since you have a discontinuous limit here, the convergence can't possibly be uniform...

The above settles the question, but you may also want (for educational purposes) to show directly that it's not possible for a given (small enough) quantity e>0 to find an N such that fn(x) would be within e of its limit for any x whenever n>N.

This is so here because, for 0<e<1, any x in [e1/n,1[ is such that fn(x) exceeds e.  However, the convergence is uniform within any interval strictly smaller than [0,1[.


brentw (Brent Watts of Hickory, NC. 2001-04-14)
[...] Explain the concept of Darboux integrals.

Before Lebesgue took a radically different  (and better)  approach to the problem of defining integrals, there were a succession of definitions which all involved dividing an interval of integration [a,b] into a finite number of segments with extremities xk such that  a=x0<x1< ... <xn=b  while making sure that every length (xk+1-xk) does not exceed a given positive quantity  e  (which can be arbtrarily small).

A certain finite sum is then computed for a given function f (see below for the main types of such sums), if the sum has a limit as e tends to zero, regardless of the chosen subdivisions, then the function f is said to be integrable and the limit is said to be its integral (in the sense of Cauchy, Riemann, Darboux, etc.).

Historically, such definitions have been based on several types of sums, including:

  • Cauchy: å (xk+1-xk) f(xk)   [This definition is now obsolete]
  • Riemann: å (xk+1-xk) f(sk)
    where sk may be anywhere between xk and xk+1
  • Darboux (lower): å (xk+1-xk) Lk
    where Lk is the greatest lower bound of f(x) for x in [xk+1-xk]
  • Darboux (upper): å (xk+1-xk) Uk
    where Uk is the least upper bound of f(x) for x in [xk+1-xk]

The last two sums correspond to the lower and upper Darboux integrals.  The nice thing is that a function f is Riemann-integrable  if and only if  its lower Darboux integral equals its upper Darboux integral.

The Riemann Integral was introduced by  Bernhard Riemann (1826-1866) in his  Habilitationsschrift  (1854)


(2014-08-09)   Measure Theory  &  Lebesgue Integrals
Theory of integration.

The foundations of the modern theory of integration were laid down in the 1902 doctoral dissertation of  Henri Lebesgue (1875-1941).

Lebesgue realized that slicing the "area" delimited by a function into horizontal slices rather than vertical ones would lead to a notion of integral that is far more satisfying than all  previous attempts.

The caveat is that the  Lebesgue Integral  requires a careful definition of the "measure" of such horizontal slices, which may be quite intricate...

 Come back later, we're
 still working on this one...

Lebesgue integration   |   Lebesgue measure   |   Lebesgue-Stieltjes integration   |  


 Joseph Fourier 
 (1768-1830)  Leonhard Euler 
 (1707-1783) brentw (Brent Watts of Hickory, NC. 2000-11-25)
How do I evaluate the Fourier series of the function
f(x)= x(2p-x) in the interval 0 < x < 2p ?

If a function   f (x)  =  ½ [ f (x-) + f (x+)]   has period  2p,  then its  Fourier expansion  is defined via:
 
 f (x)
 

=
  ao    +  
¥
å
n=1
    
[ an cos(nx)  +  bn sin(nx) ]
 
vinculum +

=
2 +
The coefficients  an  and  bn  are  twice  the average values of  cos(nx) f (x)  and  sin(nx) f (x).  They're given by Euler's formulas :

an
 

=
1  
ò  2p 
 
0
 
 f (x) cos(nx) dx
 
vinculum

=
p

bn
 

=
1  
ò  2p 
 
0
 
 f (x) sin(nx) dx
 
vinculum

=
p

For an even function, like the one at hand, the b-coefficients are all zero and we are only concerned with the first formula, giving the a-coefficients. (Conversely, the a-coefficients would all be zero for an odd function.)

In the case at hand, we  integrate by parts  twice over the interval  [0,2p]  when  n  is nonzero  (for  n = 0  we just integrate a quadratic function).  Thus,  an  is  -4/n2  if  n ¹ 0,  whereas  ao  is  4p2/3.  All told, we obtain:

 
x (2p-x)
 

=
2p2  
 4 
 
 
¥
å
n=1
   cos(nx)            [ For x between 0 and 2p ]
vinculum - vinculum

=
3 n2

The Basel Problem :

For  x = 0,  the above may serve as a proof of the famous result at right,  obtained by Euler in 1735:  The sum of the reciprocals of all nonzero perfect squares is equal to  p2/6   p2
=
¥
å
n=1
   1
vinculum vinculum
6
=
n2

 Coat-of-Arms of the 
 BERNOULLI Family 
 (Basel, Switzerland) The problem of finding the  exact  value of that sum was posed by Pietro Mengoli in 1644.  It was once known as the Basel Problem, after the hometown of Jacob Bernoulli, who was first in a long list of notorious mathematicians  (including Leibniz)  who failed to discover the above solution.  Euler first worked it out numerically to 20 decimal places  (earlier in 1735)  using the  Euler-Maclaurin formula.   He identified the value as  p2/6  before he could justify that.


p3
=
¥
å
n=0
   (-1)n  
vinculum vinculum
32
=
(2n+1)3
 
brentw (Brent Watts of Hickory, NC. 2001-03-05)
How does one prove [this relation]?
 The function f(x) is +1 or -1 when 
 x is an odd multiple of pi/2.

Consider the odd function f(x) of period 2p with an axis of symmetry at x=p/2 and equal to 2x/p when x is between 0 and p/2 (so that f(p/2)=1). Its Fourier expansion may be obtained with Euler's formulas:
 
f(x) 
 

=
8  
¥
å
n=0
   (-1)n  
  sin(2n+1)x
 
vinculum vinculum

=
p2 (2n+1)2
Integrate that to obtain the expansion of a primitive g(x) of f(x), namely:
 
g(x) 
 

=
 
 C 
 
8  
¥
å
n=0
   (-1)n  
  cos(2n+1)x
 
- vinculum vinculum

=
p2 (2n+1)3

The constant C is equal to the average of g(x) over one complete period. It depends on which value we choose for g(0). With g(0)=0, we have g(x)=x2/p for x between 0 and p/2. Because of the symmetry about x=p/2, the average C is clearly C=g(p/2)=p/4. Plug this value of C in the above relation at point x=0 (where g(x)=0 and cos((2n+1)x)=1), and you do obtain the value p3/32 for the sum you were trying to evaluate.


Consider the following expression, which generalizes the above. We are only concerned with integral (positive) values of k, but it should be noted that the function b, which is called Dirichlet's Beta Function, may be defined by analytic continuation over the entire complex plane. It has no singularities.

 
b(k)
 

=
¥
å
n=0
   (-1)n
vinculum

=
(2n+1)k

The above shows that b(3) = p3/32. Differentiating f(x), instead of integrating it, would have given b(1) = p/4, a result which is commonly obtained by computing the value of the arctangent function at x=1, using its Taylor expansion about 0.

It's worth noticing that the above method may be carried further with repeated integrations. Every other time, such an integration gives an exact expression for the alternating sum of some new power of the reciprocals of odd integers. In other words, we obtain the value of b(k) for any odd k, and it happens to be a rational multiple of p k:

b(1)  =  p/4 The general expression is
 
b(2n+1)
 

=
   (p/2) 2n+1  
  | E2n |
 
vinculum

=
2(2n)!
b(3)  =  p3/32
b(5)  =  5p5/1536
b(7)  =  61p7/184320
b(9)  =  277p9/8257536
b(11)  =  50521p11/14863564800

In this, | E2n | is an integer.  The Euler number En is the coefficient of zn/n! in the Taylor expansion of 1/ch(z)  [where ch is the hyperbolic cosine function; ch(z) = (e+e-z )/2].  Starting with n = 0, the sequence of Euler numbers is:

1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0, 2702765, 0, -199360981, 0, 19391512145, 0, -2404879675441, 0, 370371188237525, 0, ... (A122045)

We may also consider the secant function  1/cos(z)  which has the same expansion as  1/ch(z)  except that all the coefficients are positive, so that:

¼ px  /  cos(½ px)   =     å n   b(2n+1)  x 2n+1

There does not seem to be any similar expression for even powers.  In fact, b(2) is currently defined as an independent fundamental mathematical constant, the so-called Catalan Constant: G = 0.915965594177219015...

This is the exact opposite of the situation for nonalternating sums, where even powers correspond to an exact expression in terms of a rational multiple of the matching power of p, whereas odd powers do not  [that's an open conjecture].


brentw (Brent Watts of Hickory, NC. 2000-11-21)
How do you prove the following relation?
 
(p2/ab) coth(pa) coth(pb)
 

=
¥
å
m =
   
¥
å
n =
   1
vinculum

=
(m2 + a2 ) (n2 + b2 )

The relation   åm ån u(m) v(n)  =  [ åmu(m) ]   [ ånv(n) ]   holds whenever the series involved are absolutely convergent (which is clearly the case here). Therefore, we only have to establish the following simpler equality:

 
(p/a) coth(pa)
 

=
¥
å
m =
       1
vinculum

=
m2 + a2

The sum on the right-hand side looks like a series of Fourier coefficients. For what periodic function? Well, it's not difficult to see that the correct denominator is obtained for the continuous even function of period 2p which equals cosh(ax) if x is in the interval [-p,p]. When x is in that interval, the Fourier expansion may be written in two equivalent form (defining a-m to be equal to a):

 
cosh(ax)
 

=
  ao    +  
¥
å
m = 1
    
am cos(mx)   =   ½
 
¥
å
m =
 
am cos(mx)
 
vinculum +

=
2 +

With Euler's formula, you obtain am = [2a(-1)m/p] sinh(pa) / (m2 + a2 ). At the point x = p, we have cos(mx)= (-1)m and the above relation thus translates into the desired equality [just divide both sides of the relation by (a/p) sinh(pa)].


brentw (Brent Watts of Hickory, NC. 2001-04-14)
How do you use the Fourier series for the function f(x) = ex for x in ]0,2p[ to find the sum [S] of the series 1/(k2 + 1) ? [ k=1 to ¥ ]

Use Euler's formulas to compute the Fourier coefficients of f(x). Note that if you consider f as a function of period 2p equal to exp(x) in ]0,2p[, it has a jump discontinuity at any point x=2np (where n is any integer). This means (and it's important for the rest) that the Fourier series converge to the half-sum of the left limit and the right limit at such points of discontinuity, in particular the value at point 0 is [exp(2p)+1]/2.

Now, the computation of the Fourier coefficients is easy if you notice that exp(x)cos(kx) and exp(x)sin(kx) are the real and imaginary parts of exp((1+ki)x) (it's clear we will only need the real part, but I'll pretend I did not notice). The indefinite integral of that is simply exp((1+ki)x)/(1+ki), which we may also express as exp((1+ki)x) (1-ki)/(1+k2). The definite integral from 0 to 2p is thus (exp(2p)-1)(1-ki)/(1+k2), and the Fourier coefficients are obtained by multiplying this by 1/p and using the real and imaginary part separately. All told:

f(x) = [exp(2p)-1]/p (½ + SUM[ k=1 to ¥, (cos(kx) - k sin(kx))/(1+k2) ] )

All you have to do is apply this to x=0 (this is why we did not really need the coefficients of sin(kx)). With the above remark to the effect that the LHS really represents [f(x-)+f(x+)]/2 at any jump discontinuity like x=0, we obtain:

[exp(2p)+1] / 2 = [exp(2p)-1] / p ( ½ + S )

where S is the sum you were after.  Therefore:

S = p/2 - ½ + p / [ exp(2p)-1 ] = 1.076674047...

This is also a special case (a = 1) of the relation established above in the form:

p coth(p)   =   1 + 2 S


brentw (Brent Watts of Hickory, NC. 2000-11-28)
[...] Please explain the Gibbs phenomenon of Fourier series.

At a point x where a function f has a jump discontinuity, any partial sum of its Fourier series adds up to a function that has an "overshoot" (i.e., a dampened oscillation) whose initial amplitude is about 9% of the value of the jump J=|f(x+)-f(x-)|.

This amplitude is not reduced by adding more terms of the Fourier series. It's not difficult to prove that, with n terms, the maximum value of the overshoot occurs at/near a distance of p/2n on either side of x. (You may do the computation with any convenient function having a jump J; I suggest f(x)=sign(x)J/2 between -p and p. Adding a continuous function to that would put you back to the "general" case without changing the nature or amplitude of the Gibbs oscillations.)

When n tends to infinity, the maximum reached by the first overshoot oscillation is about 8.948987% of the jump J. This value is precisely (2G/p-1)/2, where G is known as the Wilbraham-Gibbs Constant:

G  =   ó p
õ
0  
sin(q)/q dq  
=   1.8519370519824661703610533701579913633458...

This is sometimes called "the 9% overshoot", as it is about 9% of the total jump J.  [It's 18% (17.89797...%) when half the jump (J/2) is used as a unit.]

This tells you exactly what kind of convergence is expected from a Fourier series about a discontinuity of  f. For a small h, you can always increase the number of Fourier terms so that Gibbs oscillations are mostly confined to the very beginning of the interval [x,x+h].  This is somewhat like the convergence to zero of the sequence of functions f(n,x) defined as being equal to 4nx(1-nx) for x between 0 and 1/n, and zero elsewhere. f(n,x) always reaches a maximum value of 1 for x=1/2n. f(n,x) does converge to zero, but it is not uniform convergence! Same thing with partial Fourier sums in a neighborhood of a jump discontinuity...


ó  b
 f (x) dx
PV
õ
a
brentw (Brent Watts of Hickory, NC. 2000-12-08)
What is the Cauchy principal value (PV) of an integral?

If f has no singularities, the principal value (PV) is just the ordinary integral.

 Augustin Cauchy 
 (1789-1857)

If the function f has a single singularity q between a and b (a<b), the Cauchy principal value of its integral from a to b is the limit (whenever it exists), as e tends to 0+, of the sum of the integral from a to q-e and the integral from q+e to b.  Also, if the interval of integration is ]-¥,+¥[ with a singularity at ¥ the principal value is the limit, whenever its exists, of the integral in the interval ]-A,+A[ as A tends to infinity.

When f has a discrete number of singularities between a and b (a and b excluded, unless both are infinite), the PV of its integral may be obtained by splitting the interval [a,b] into a sequence of intervals each containing a single singularity.  The above applies to each of these, and the PV of the integral over the entire interval is obtained by adding the principal values over all such subintervals.

The fact that the principal value is used may be indicated by the letters PV before the integral sign, or by crossing with a small horizontal dash the integral sign (see illustration above). However, it is more or less universally understood that the Cauchy principal value is used whenever needed, and some authors don't bother to insist on this with special typography.


brentw (Brent Watts of Hickory, NC. 2000-11-21)
How do I solve the differential equation 2(1-x)y" + (1+x)y' + [x - 3 - (x-1)2exp(x)]y = 0 about the pole x=1?

The singularity at  x = 1  is a  regular  one  (which means that if the coefficient of y" is normalized to 1, the coefficient of y' has at most a single pole at x=1 and the coefficient of y has at most a double pole at x=1).  Therefore, the method of Frobenius is applicable.  It consists in finding a solution in the form of a so-called Frobenius series of the following form (where h=x-x0 in general, and h=x-1 here) with a(0) nonzero:

y = hm [ a(0) + a(1) h + a(2) h2 + a(3) h3 + ... ]

In the above, m is not necessarily an integer, so that a Frobenius series is more general than either a Taylor series (for which m is a natural integer) or a Laurent series (for which m is any integer). In the DE we're asked to study, we have:

-2h y" + (2+h) y' + [h-2-h2exp(1+h)] y = 0

The method of Frobenius is simply to expand the above LHS in terms of powers of h to obtain a sequence of equations that will successively give the values of a=a(0), b=a(1), c=a(2), d=a(3), etc. Let's do it. The above LHS is hm-1 multiplied by:

[-2am(m-2)] +[a(m-2)-2b(m2-1)]h +[a+b(m-1)-2cm(m+2)]h2 +O(h3)

We have to successively equate to zero all the square brackets.  Since a is nonzero, the first square bracket gives us the acceptable value(s) of the index m (this is a general feature of the method and this first critical equation is called the indicial equation).  Generally, the indicial equation has two roots (for a second-degree DE) and this gives you a pair of independent solutions.  Usually, when the roots differ by an integral value (like here) you've got (somewhat) bad news, since the Frobenius method is only guaranteed to work for the "larger" of the two values of m.  However, "by accident" you're in luck here:

The case m=0 gives b=a (second bracket). Then, the third bracket gives zero for the coefficient of c  (that's the usual problem you encounter after N steps when starting with the smaller root, if the two roots differ by N)  but it so happens that the rest of the bracket is zero too!  (That's exceptional!) So you can continue with an arbitrary value of c and obtain d as a linear combination of a and c using the next bracket (which I was too lazy to work out, since I knew  tough  problems could not occur past that point).

The way to proceed from here is to first use a=1 and c=0 to get the first solution as a Frobenius series F(h), then a=0 and c=1 to get a linearly independent solution G(h). The general solution about the singularity x=1 is then  a F(x-1) + c G(x-1). (You don't have to bother with the index m=2 in this particular case.)


brentw (Brent Watts of Hickory, NC. 2000-11-21)   Laurent Series
How to determine the Laurent series of a function about a singular point.

For each singular point  (or pole)  zo, you want to expand  f (zo+h).  If a pole has multiplicity  n,  then hn  f (zo+h) is normally an analytic function.  Compute its Taylor expansion about h=0 and divide that series by hn to have the  Laurent series  about that particular pole.

Let's present the computation step-by-step for the following simple example:

f (z)   =   1 / [z (z-1)2 ]

Let's examine the double pole at  z = 1.

First we compute   f (1+h)   [in the neighborhood of the pole, h is  small ].  It's just a matter of replacing  z  by  (1+h).  Nothing to it:

f (1+h)   =   1 / [ (1+h)((1+h)-1)2 ]   =   1 / [ (1+h)h2 ]

Multiply this by  h to obtain an analytic function about  h = 0,  namely:

g(h)   =   h2  f (1+h)   =   1 / (1+h)

The Taylor expansion of  g  is well known:

g(h)   =   1 - h + h2 - h3 + h4 - h5 + h6 - h7 +   ...

Since  f (1+h) = g(h)/h2, we divide the above by  h2  to obtain the Laurent expansion of  f (1+h)  about h=0 or, equivalently, of  f (z)  about z=1 :

f (1+h) = 1/h2 - 1/h + 1 - h + h2 - h3 + h4 - h5 +   ...

Usually, we're only concerned with the coefficient of 1/h, which is called the residue for that pole (here it's equal to -1).  The integral of a function along any closed contour encircling a certain number of poles is equal to  2pi  times the sum of the residues for those poles. This can be a practical way to compute easily many definite integrals that would otherwise be difficult to obtain.  Examples follow.

Many textbook insist on the following format  (which I don't recommend):

f (z)   =   1/(z-1)2 - 1/(z-1) + 1 - (z-1) + (z-1)2 - (z-1)3 +   ...

It's sometimes desirable to present the function globally as the sum of so-called  simple elements  about every pole and an  entire  function  (which is just a polynomial when the function is  rational, like this one).

f (z)   =   1/(z-1)2  -  1/(z-1)  +  1/z  +  0

That type of reduction allows the immediate integration of rational functions:

ò   f (z) dz   =   -1/(z-1)  -  ln(z-1)  +  ln(z)


ò  ¥   dx
vinculum
 
0
(1+x2 )Öx
yourm0mz ( 2001-12-15)
How do you find this definite integral?
(I am using the positive x-axis as a branch cut.)

When attempting to apply Cauchy's residue theorem  Augustin Cauchy 
 (1789-1857) [the fundamental theorem of complex analysis] to multivalued functions (like the square root function involved here), it is important to specify a so-called "cut" in the complex plane were the function is allowed to be discontinuous, so that it is everywhere else continuous and single-valued.

In the case of the square-root function, it is not possible to give a continuous definition valid around any path encircling the origin. Therefore, a so-called "branch-cut" line must be specified which goes from the origin to infinity. The usual choice is [indeed] to use the positive x-axis for that purpose. This choice means that, when the angle q is in [0,2p[, the "square root" of the complex number z = r exp(iq) is simply Öz º Ör exp(iq/2)  (the notation Ör being unambiguous because r is a positive real number and its square root is thus defined as the only positive real number of which it is the square). This definition does present a discontinuity when crossing the positive real axis (a difficulty avoided only with the introduction of Riemann surfaces, which are beyond the scope of our current discussion).

Path With the above definition of the square root of a complex argument, we may thus apply the Residue Theorem to the function f(z)=1/(1+z2)Öz on any contour which does not cross the positive real axis. We may choose the contour pictured at left, which does not encircle the origin [this would be a no-no, regardless of the chosen "branch cut"] but encloses the pole at +i when the outer circle is big enough.

On the outer semicircle, the quantity |f(z)| eventually becomes much smaller than the reciprocal of the length of the path of integration. Therefore, the contribution of the outer semicircle to the contour integral tends to zero as the radius tends to infinity. The smaller semicircle is introduced to avoid the singularity at the origin, but its contribution to the contour integral is infinitely small when its radius is infinitely small. What remains, therefore, is the contribution of the two straight parts of the contour. The integral along the right part is exactly the integral we are asked to compute, whereas the left part contributes i times that quantity. All told, the limit of the contour integral is (1+i) times the integral we seek.

Cauchy's Theorem states that the contour integral equals 2pi times the sum of the residues it encircles.

In this case, there's only one such residue, at the pole i. The value of the residue at pole i  is the limit as h tends to zero of f(i+h), namely 1/2iÖi, so the value of the contour integral is i = 2(1+i)/2. As stated above, this is (1+i) times the integral we want. Therefore, the value of that integral is exactly p/Ö2, or about 2.221441469...


ò  ¥   xa dx
vinculum
 
0
1 + x2
For what values of a does this integral converge?
What's the value of the integral when it converges?

The previous article deals with the special case a = -1/2.
In general, we see that the integral makes sense in the neighborhood of zero if a>-1 and it converges in the neighborhood of when a<1. All told, the integral converges when a is in the open interval ]-1,1[.

Path

We apply the above method to f(z) = za/(1+z2) [defining za with the positive x-axis as branch cut] on the same contour (pictured at right). The smaller semicircle is useless when a is positive and it has a vanishing contribution otherwise (when a>-1). The contribution of the outer semicircle is vanishingly small also (when a<1) because |f(z)| multiplied by the length of the semicircle becomes vanishingly small when the radius becomes large enough. On the other hand, the contribution of the entire positive x-axis is the integral we are after, whereas the negative part of the axis contributes exp(ipa) as much. All told therefore, Cauchy's theorem tells us that our integral is 2pi/(1+exp(iap)) times the residue of  f at the pole z = i.

The residue at z = i  is the limit, as h tends to zero, of f(i+h), which is simply exp(iap/2)/2i. This makes the integral equal to

p exp(iap/2) / (1+exp(iap))

That boils down to   p/(2 cos ap/2)   so that we obtain:

ó
õ
 ¥   xa dx     =     p [ -1 < a < 1 ]
vinculum vinculum
0 1 + x2 2 cos (ap/2)

We may notice that this final result is an even function of a, which we could have predicted with the simple change of variable  y = 1/x...


(2007-05-08)   The Vocabulary of Complex Analysis
Holomorphic functions, entire functions, meromorphic functions, etc.

Holomorphic Functions
(analytic functions of a complex variable)

A complex function  f  of a complex variable which is differentiable about every point  z  of its domain is  interchangeably  called an analytic function, an holomorphic function, or a  complex differentiable function.  This is to say:

lim    f (z+h)  -  f (z)     =   f ' (z)
Vinculum
h ® 0 h

The existence of such a  derivative function  ( f ' )  is a much more restrictive condition in the complex realm than among real numbers, since it implies essentially that the original function is differentiable  infinitely many times  and possesses a convergent expansion as a  Taylor series  about every point  x  inside  of its domain of definition.  It's thus essentially a weighted sum of powers of  (z-x).

 f (z)    =  
¥
å
n = 0
   
 f (n) (x)
  (z-x) n   where   f (0) = f
vinculum
n!
 
 f (n+1)  =  [ f (n) ]'

We may break down complex quantities into their real and imaginary parts:

f (z)   =   f (x+iy)   =   u + i v   =   u(x,y) + i v(x,y)

In this, both  u  and  v  are real functions of the  two  real variables  x  and  y.  The above differentiability of  f  as a function of the complex variable  x+iy  is equivalent to the following equation in terms of the partial derivatives of  f  with respect to the two real variables  x  and  y.

i f / x   =   f / y

In terms of  u  and  v,  this translates into the following differential equations, known as the  Cauchy-Riemann equations,  which are a necessary and sufficient condition for the related function  f  to be holomorphic  (note how these equations imply that both u and v are necessarily differentiable infinitely many times with respect to x and y as soon as they are known to be differentiable once).

u / x   =   v / y     and     v / x   =   - ¶u / y

Entire Functions and Liouville's Theorem

An  holomorphic function  defined over the whole complex plane  (without any singularities)  is called an  entire functionLiouville's Boundedness Theorem  asserts that such a function can only be bounded if it's  constant.

Meromorphic Functions

meromorphic function  f  over some domain D is an  holomorphic function  on the domain obtained by removing from D a discrete set of isolated ordinary  poles  (as opposed to  essential singularities).  That's to say that a neighborhood of any such singularity  x  exists where the function  f (z)  multiplied by some power of  (z-x)  is simply an holomorphic function (i.e., without singularities).

Etymologically, meros (meroV = part) is opposed to holos (holoV = whole).

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