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.

X set, K c P(X), W c P(X), K and W are monotone, A c X

What can we say about { B:-K : AuB:-W } ?

What can we say about { B:-K : AuB:-W } ?

It is monotone.

X set, K c P(X), W c P(X), K and W are monotone, A c X

What can we say about { B:-K : A\B:-W } ?

What can we say about { B:-K : A\B:-W } ?

It is monotone.

X set, K c P(X), W c P(X), K and W are monotone, A c X

What can we say about { B:-K : AnB:-W } ?

What can we say about { B:-K : AnB:-W } ?

It is monotone.

Let z be a complex number.

Re(i*z) = ???

Re(i*z) = ???

Re(i*z) = (-1)*Im(z)

Let z be a complex number.

Im(i*z) = ???

Im(i*z) = ???

Im(i*z) = Re(z)

Let z be a complex number.

| Re(z) | <= ???

| Re(z) | <= ???

| Re(z) | <= |z|

Let z be a complex number.

| Im(z) | <= ???

| Im(z) | <= ???

| Im(z) | <= |z|

Let a[n] be a sequence of complex numbers, indexed from n=0 to
n=oo.

Suppose lim_sup |a[n]|^(1/n) <= 1.

Let S[n] = a[0] + a[1] + ... + a[n].

Let z be a complex number with |z|<1.

What can we say about +(n=0 to n=oo) [ S[n]*z^n ] ?

Suppose lim_sup |a[n]|^(1/n) <= 1.

Let S[n] = a[0] + a[1] + ... + a[n].

Let z be a complex number with |z|<1.

What can we say about +(n=0 to n=oo) [ S[n]*z^n ] ?

+(n=0 to n=oo) [ S[n]*z^n ] = 1/(1-z) * ( +(n=0 to n=oo) [
a[n]*z^n ] )

Hint: use the theorem about multiplying series.

page 14 in OLDTIMER

Hint: use the theorem about multiplying series.

page 14 in OLDTIMER

Let A be an connected subset of |R.

Let g : A -> |C diffable.

Suppose that g'(x)=0 for all x:-A.

What can we say about function g?

Let g : A -> |C diffable.

Suppose that g'(x)=0 for all x:-A.

What can we say about function g?

It is constant.

page 29 in OLDTIMER

page 29 in OLDTIMER

Let D be an open connected subset of |C.

Let g : D -> |C diffable.

Suppose that g'(z)=0 for all z:-D.

What can we say about function g?

Let g : D -> |C diffable.

Suppose that g'(z)=0 for all z:-D.

What can we say about function g?

It is constant.

page 30 in OLDTIMER

page 30 in OLDTIMER