Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

Let G be a subset of |C. Let f : G -> |C. Let a:-|C.

What is the verbal description of this situation:

There exists a number r > 0 and function g : B(a;r) -> |C such that:

(1) a doesn't belong to G

(2) B(a;r) \ {a} c G

(3) g is diffable

What is the verbal description of this situation:

There exists a number r > 0 and function g : B(a;r) -> |C such that:

(1) a doesn't belong to G

(2) B(a;r) \ {a} c G

(3) g is diffable

function f has a removable isolated singularity at a

Let G be an open subset of a metric space X.

Let A c X. Suppose that G n Clo(A) is non-empty.

What can we conclude from this?

Let A c X. Suppose that G n Clo(A) is non-empty.

What can we conclude from this?

G n A is non-empty

Let G be an open subset of a metric space X.

Let A c X. Suppose that G n A is empty.

What can we conclude from this?

Let A c X. Suppose that G n A is empty.

What can we conclude from this?

G n Clo(A) is empty

Let G,A be subsets of a metric space X.

Suppose that G n A is empty and G n Clo(A) is non-empty.

What can we conclude from this?

Suppose that G n A is empty and G n Clo(A) is non-empty.

What can we conclude from this?

G is not open

If x:-|R, then let [x] be the integer, x-1 < [x] <= x.

Prove that /\a,b:-|R [a] + [b] <= ???.

Prove that /\a,b:-|R [a] + [b] <= ???.

/\a,b:-|R [a] + [b] <= [a + b]

page 120 in OLDTIMER

page 120 in OLDTIMER

(1) 0 <= a[n] <= oo

(2) 0 <= b[n] <= oo,

(3) lim a[n] = a, 0 < a <= oo

(4) lim b[n] = b, 0 < b <= oo

What can we conclude from this?

(2) 0 <= b[n] <= oo,

(3) lim a[n] = a, 0 < a <= oo

(4) lim b[n] = b, 0 < b <= oo

What can we conclude from this?

lim a[n]*b[n] = a*b

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

Suppose that f has an isolated singularity at a.

(1) f has a removable singularity at a

(2) ??????

Recall an equivalent condition without proof.

Suppose that f has an isolated singularity at a.

(1) f has a removable singularity at a

(2) ??????

Recall an equivalent condition without proof.

(2) lim(z->a) (z-a)*f(z) = 0

MRW hasn't done this proof yet. 2001.02.27; 2001.05.26

(page 103, Conway, "Functions of One Complex Variable", chapter V)

MRW hasn't done this proof yet. 2001.02.27; 2001.05.26

(page 103, Conway, "Functions of One Complex Variable", chapter V)

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

What does it mean that f has a pole at b?

What does it mean that f has a pole at b?

(1) b doesn't belong to G

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

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

or shorter:

(1) f has an isolated singularity at b

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

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

or shorter:

(1) f has an isolated singularity at b

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

(1) b doesn't belong to G

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

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

What is the verbal description of this situation?

(1) b doesn't belong to G

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

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

What is the verbal description of this situation?

function f has a pole at point b

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

Suppose that function f has a removable isolated singularity at point b.

Can it also have a pole at b?

Suppose that function f has a removable isolated singularity at point b.

Can it also have a pole at b?

NO.

By virtue of our supposition there exists a number r > 0

and a function g : B(b;r) -> |C such that:

(1) b doesn't belong to G

(2) B(b;r) \ {b} c G

(3) g is diffable

(4) g=f on B(b;r)\{b}

By virtue of our supposition there exists a number r > 0

and a function g : B(b;r) -> |C such that:

(1) b doesn't belong to G

(2) B(b;r) \ {b} c G

(3) g is diffable

(4) g=f on B(b;r)\{b}