COMPLEX ANALYSIS: Maple Worksheets, 2009
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

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COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
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CHAPTER 7 TAYLOR and LAURENT SERIES

Section 7.4 Singularities, Zeros and Poles

The point is called a singular point , or singularity , of the complex function if is not analytic at the point , but every neighborhood of contains at least one point at which is analytic. For example, the function is not analytic at , but is analytic for all other values of . Thus, the point is a singular point of . As another example, consider the function . We saw in Section 5.2 that is analytic for all except at the origin and at the points on the negative real axis. Thus, the origin and each point on the negative real axis is a singularity of .

The point is called an isolated singularity of a complex function if is not analytic at , but there exists a real number such that is analytic everywhere in the punctured disk . The function has an isolated singularity at .

The function , however, has a singularity at (or at any point of the negative real axis) that is not isolated, because any neighborhood of will contain points on the negative real axis, and is not analytic at those points. Functions with isolated singularities have a Laurent series because the punctured disk is the same as the annulus . We now look at this special case of Laurent's theorem in order to classify three types of isolated singularities.

Example for a pole, Page 291. Consider the function .
The leading term in the series expansion is and
in the same manner as .

> f:='f': L:='L': s:='s': S:='S': z:='z': Z:='Z':
f := z -> cot(z):
S := series(f(Z), Z=0, 8):
s := convert(S, polynom):
LS := z -> subs(Z=z,s):
`f(z) ` = f(z);
`f(z) ` = subs(Z=z,S);
L[7](z) = LS(z);

> x:='x': y:='y':
plot({f(x),LS(x)}, x=-3.14..3.14, y=-15..15,
title=`y = cot(x) and y = L7(x)`,
tickmarks=[7,7]);

Example of a removable singularity, Page 291. The function
has a removable singularity at .

> f:='f': L:='L': s:='s': S:='S': z:='z': Z:='Z':
f := z -> sin(z)/z:
`f(z) ` = f(z);
S := series(sin(Z), Z=0, 11)/Z:
`f(z) ` = subs(Z=z,S);
S := series(f(Z), Z=0, 11):
`f(z) ` = subs(Z=z,S);
s := convert(S, polynom):
LS := z -> subs(Z=z,s):
L[8](z) = LS(z);

Example of a removable singularity, Page 291. The function
has a removable singularity at .

> f:='f': L:='L': s:='s': S:='S': z:='z': Z:='Z':
f := z -> (cos(z) -1)/z^2:
`f(z) ` = f(z);
S := series((cos(Z) -1), Z=0, 11)/Z^2:
`f(z) ` = subs(Z=z,S);
S := series(f(Z), Z=0, 11):
`f(z) ` = subs(Z=z,S);
s := convert(S, polynom):
LS := z -> subs(Z=z,s):
`f(z) ` = f(z);
L[8](z) = LS(z);

Example of a pole of order 2, Page 291. The function
has a pole of order at .

> f:='f': L:='L': s:='s': S:='S': z:='z': Z:='Z':
f := z -> sin(z)/z^3:
`f(z) ` = f(z);
S := series(sin(Z), Z=0, 13)/Z^3:
`f(z) ` = subs(Z=z,S);
S := series(f(Z), Z=0, 13):
`f(z) ` = subs(Z=z,S);
s := convert(S, polynom):
LS := z -> subs(Z=z,s):
L[8](z) = LS(z);

Example of a simple pole, Page 291. The function
has a simple pole at .

> f:='f': L:='L': s:='s': S:='S': z:='z': Z:='Z':
f := z -> exp(z)/z:
`f(z) ` = f(z);
S := series(exp(Z), Z=0, 8)/Z:
`f(z) ` = subs(Z=z,S);
S := series(f(Z), Z=0, 8):
`f(z) ` = subs(Z=z,S);
s := convert(S, polynom):
LS := z -> subs(Z=z,s):
L[6](z) = LS(z);

Examples of an essential singularity, Page 291. The function
has an essential singularity at .

> f:='f': L:='L': p:='p': s:='s': z:='z': Z:='Z':
f := z -> z^2*sin(1/z):
s := convert(series(sin(z), z=0, 12), polynom):
S := subs(z=1/Z,s):
p := z -> subs(Z=z, expand(Z^2 * S)):
`f(z) ` = f(z);
L[9](z) = p(z);

There will be infinitely many terms involving negative powers of ,
hence, has an essential singularity at .

Theorem 7.10 A function analytic in has a zero of order k
at the point if and only if its Taylor series given by
has and .

Example 7.10, Page 292. Show that has a zero of order at .

> f:='f': p:='p': P:='P': s:='s': z:='z': Z:='Z':
f := z -> z*sin(z^2):
s := convert(series(f(Z), Z=0, 24), polynom):
p := z -> subs(Z=z,s):
`f(z) ` = f(z);
P[23](z) = p(z);

Thus, has a zero of order at .

Theorem 7.11 Suppose is analytic in . Then has a zero of order k at the point if and only if it can be expressed in the form , where is analytic at and .

Corollary 7.4 If and are analytic at and have zeros of orders and , respectively,

then their product has a zero of order at the point .

Example 7.11, Page 293. Show that has a zero of order at .

> f:='f': p:='p': P:='P': s:='s': z:='z': Z:='Z':
f := z -> z^3*sin(z):
s := convert(series(f(Z), Z=0, 15), polynom):
p := z -> subs(Z=z,s):
`f(z) ` = f(z);
P[14](z) = p(z);

Thus, has a zero of order at .

Theorem 7.12 A function analytic in the punctured disk has a pole of order k at if and only if it can be expressed in the form

,

where is analytic at and .

Corollary 7.5 If is analytic and has a zero of order k at ,

then has a pole of order k at .

Corollary 7.6 If has a pole of order k at , then has a removable singularity at .

If we define , then has a zero of order k at .

Example 7.12, Page 295. Locate the zeros and poles of ,
and determine their order.

> h:='h': L:='L': s:='s': s:='s': z:='z': Z:='Z':
h := z -> tan(z)/z:
`h(z) ` = h(z);
S := series(h(Z), Z=0, 12):
`s(z) ` = subs(Z=z,S);
s := convert(S, polynom):
LS := z -> subs(Z=z,s):
L[10](z) = LS(z);

Thus, has a removable singularity at .

Next consider the points = ... , , ... .

> tan(pi)/pi = tan(Pi)/Pi;
tan(2*pi)/(2*pi) = tan(2*Pi)/(2*Pi);
tan(3*pi)/(3*pi) = tan(3*Pi)/(3*Pi);
tan(4*pi)/(4*pi) = tan(4*Pi)/(4*Pi);
tan(5*pi)/(5*pi) = tan(5*Pi)/(5*Pi);

Thus has simple zeros at where .

Next consider the points = ..., , ... .

> tan(pi/2)/(pi/2) = limit(h(x),x=Pi/2,left);
tan(3*pi/2)/(3*pi/2) = limit(h(x),x=3*Pi/2,left);
tan(5*pi/2)/(5*pi/2) = limit(h(x),x=5*Pi/2,left);
tan(7*pi/2)/(7*pi/2) = limit(h(x),x=7*Pi/2,left);

Thus has poles at = ..., , ... .

> s := z -> series(h(z), z=Pi/2, 3):
`h(z) ` = h(z);
`h(z) ` = s(z);

Hence has simple poles at where is an integer.

Example 7.13, Page 296. Locate the poles of ,
and specify their order.

> g:='g': z:='z':
g := z -> 1/(5*z^4 + 26*z^2 + 5):
`g(z) ` = g(z);

Find the singularities of .

> Zn := sort([solve(denom(g(z))=0, z)]):
`For g(z) ` = g(z);
`The singularities are:`;
z1 := subs(z=Zn[1],z): z[1] = z1;
z2 := subs(z=Zn[2],z): z[2] = z2;
z3 := subs(z=Zn[3],z): z[3] = z3;
z4 := subs(z=Zn[4],z): z[4] = z4;

Thus , has four simple poles.

Example 7.14, Page 296. Locate the zeros and poles of ,
and determine their order.

> g:='g': L:='L': p:='p': s:='s': S:='S': z:='z': Z:='Z':
g := z -> Pi*cot(Pi*z)/z^2:
s := series(Pi*cot(Pi*z), z=0, 7)/z^2:
S := series(g(z), z=0, 7):
p := convert(series(Pi*cot(Pi*Z), Z=0, 7), polynom):
LS := z -> subs(Z=z,expand(p/z^2)):
`g(z) ` = g(z);
`g(z) ` = s;
`g(z) ` = S;
L[3](z) = LS(z);

Thus , has a pole of order at .

Next consider the points z = ... , , ... .

> pi*cot(pi)/(pi)^2 = limit(g(x),x=1,right);
2*pi*cot(2*pi)/(2*pi)^2 = limit(g(x),x=2,right);
3*pi*cot(3*pi)/(3*pi)^2 = limit(g(x),x=3,right);
s := z -> series(g(z), z=1, 5):
`g(z) ` = g(z),` Expand about the point `,z[0] = 1;
`g(z) ` = s(z);
s := z -> series(g(z), z=2, 5):
`g(z) ` = g(z),` Expand about the point `,z[0] = 2;
`g(z) ` = s(z);

Thus has simple poles at the points z = ... , , ... . .

End of Section 7.4.