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G c |C; f : G -> |C; b:-|C

Suppose that function f has a pole at point b.

Can it also have a removable singularity at b?

Suppose that function f has a pole at point b.

Can it also have a removable singularity at b?

NO.

By virtue of our supposition:

(1) a doesn't belong to G

(2) there exists B = {z:-|C : 0 < |z-a| < r} c G such that f is diffable on B

(3) lim(z->b) |f(z)| = oo

By virtue of our supposition:

(1) a doesn't belong to G

(2) there exists B = {z:-|C : 0 < |z-a| < r} c G such that f is diffable on B

(3) lim(z->b) |f(z)| = oo

G c |C; f : G -> |C; b:-|C

What does it mean that function f has an essential isolated singularity at point b?

What does it mean that function f has an essential isolated singularity at point b?

(1) f has an isolated singularity at b

(2) the singularity is not removable

(3) the singularity is not a pole

(2) the singularity is not removable

(3) the singularity is not a pole

Consider the complex function f(z)=exp(1/z).

What kind of isolated singularity does it have at 0?

What kind of isolated singularity does it have at 0?

essential

(1) lim(n->oo) |f(1/n)| = oo

(2) lim(n->oo) |f(i/n)| = 1 < oo

(1) not removable

(2) not a pole

(1) lim(n->oo) |f(1/n)| = oo

(2) lim(n->oo) |f(i/n)| = 1 < oo

(1) not removable

(2) not a pole

Let m be a positive integer.

Consider the complex function f(z)=z^m * sin(1/z).

What kind of isolated singularity does it have at 0?

Consider the complex function f(z)=z^m * sin(1/z).

What kind of isolated singularity does it have at 0?

essential

(1) lim(n->oo) |f(i/n)| = oo

(2) lim(n->oo) |f(1/n)| = 0

(1) not removable

(2) not a pole

(1) lim(n->oo) |f(i/n)| = oo

(2) lim(n->oo) |f(1/n)| = 0

(1) not removable

(2) not a pole

Consider the complex function f(z) = sin(z)/z.

What kind of isolated singularity does it have at 0?

What kind of isolated singularity does it have at 0?

removable

====================

You can prove your answer by referring to the definition only.

This is also an opportunity to test the unproven theorem. (MRW 2001.02.27)

THEOREM: G c |C; f : G -> |C; b:-|C

thesis: If function f has an isolated singularity at poin

====================

You can prove your answer by referring to the definition only.

This is also an opportunity to test the unproven theorem. (MRW 2001.02.27)

THEOREM: G c |C; f : G -> |C; b:-|C

thesis: If function f has an isolated singularity at poin

Consider the complex function f(z) = ( cos(z) - 1 ) / z.

What kind of isolated singularity does it have at 0?

What kind of isolated singularity does it have at 0?

removable

====================

You can prove your answer by referring to the definition only.

This is an opportunity to test the unproven theorem. (MRW 2001.02.04)

THEOREM: G c |C; f : G -> |C; b:-|C

thesis: If function f has an isolated singularity at poin

====================

You can prove your answer by referring to the definition only.

This is an opportunity to test the unproven theorem. (MRW 2001.02.04)

THEOREM: G c |C; f : G -> |C; b:-|C

thesis: If function f has an isolated singularity at poin

lim(x->0,y->0) (x-sin(y))*y^2 / (y^2 + sin(x-y)*sin(x-y) ) = ???

= 0

hint: y^2 <= y^2 + sin(x-y)*sin(x-y)

hint: y^2 <= y^2 + sin(x-y)*sin(x-y)

lim(x->0) x*sin(x) / ( x^2 + sin(x)*sin(x) ) = ???

= 1/2

hint: use the squeeze theorem

hint: use the squeeze theorem

Consider the complex function f(z) = (z+1)/z.

Does it have a removable isolated singularity at 0 ?

Justify your answer in two different ways.

Does it have a removable isolated singularity at 0 ?

Justify your answer in two different ways.

NO. It actually has a pole. Notice that lim(z->0) |f(z)| = oo.

---------------------------------

(1) lim(z->0) z*f(z) = 1

hence not (2) lim(z->0) (z-0)*f(z) = 0.

And (2) is a necessary condition for a removable isolated singularity.

(page 103, Conway, "Functions of One Complex Variable", cha

---------------------------------

(1) lim(z->0) z*f(z) = 1

hence not (2) lim(z->0) (z-0)*f(z) = 0.

And (2) is a necessary condition for a removable isolated singularity.

(page 103, Conway, "Functions of One Complex Variable", cha

w:-|C

2*sin^2(w) = ???

2*sin^2(w) = ???

2*sin^2(w) = 1 - cos(2*w)