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 A be an open subset of |R. Let h : A -> |C.

Suppose that h is diffable.

What can we say about function Im(h)?

Suppose that h is diffable.

What can we say about function Im(h)?

It is diffable and [Im(h)]' = Im(h').

page 46 in OLDTIMER

page 46 in OLDTIMER

Let D be an open subset of |C.

Let a,b :- |C and the interval [a,b] c D.

Let f : D -> |C diffable.

We have /\z:-D |f'(z)|<=M.

Prove that |f(b)-f(a)| <= ???.

Let a,b :- |C and the interval [a,b] c D.

Let f : D -> |C diffable.

We have /\z:-D |f'(z)|<=M.

Prove that |f(b)-f(a)| <= ???.

|f(b)-f(a)| <= 2*M*|b-a|

page 47 in OLDTIMER

page 47 in OLDTIMER

Let M be an open connected subset of |C.

Let h : M -> |C be diffable.

Suppose that Re(h) is constant on M.

What can we say about function h?

Let h : M -> |C be diffable.

Suppose that Re(h) is constant on M.

What can we say about function h?

It is constant.

page 49 in OLDTIMER

page 49 in OLDTIMER

Let M be an open connected subset of |C.

Let h : M -> |C be diffable.

Suppose that Im(h) is constant on M.

What can we say about function h?

Let h : M -> |C be diffable.

Suppose that Im(h) is constant on M.

What can we say about function h?

It is constant.

page 49 in OLDTIMER

page 49 in OLDTIMER

Let D be an open subset of |R^2. Let (a,b):-D. Let f : D -> |R.

(1) the second partial derivatives fxy and fyx exist on D

(2) fxy and fyx are continuous at (a,b).

Is it true that fxy(a,b) = fyx(a,b)?

(1) the second partial derivatives fxy and fyx exist on D

(2) fxy and fyx are continuous at (a,b).

Is it true that fxy(a,b) = fyx(a,b)?

Yes.

page 54 in OLDTIMER

page 54 in OLDTIMER

Let a,b:-|R. Let f : [a,b] -> |C.

Suppose that Re(f) and Im(f) are Riemann-integrable on [a,b].

Define the complex integral of f on [a,b].

Suppose that Re(f) and Im(f) are Riemann-integrable on [a,b].

Define the complex integral of f on [a,b].

Integral([a,b],f) := Integral([a,b],Re(f)) +
i*Integral([a,b],Im(f))

Let a,b:-|R. Let f : [a,b] -> |C. Let f be integrable.

Prove that

/\w:-|C Integral([a,b],w*f) = w * Integral([a,b],f)

Prove that

/\w:-|C Integral([a,b],w*f) = w * Integral([a,b],f)

page 56 in OLDTIMER

Think about complexification of real vector spaces.

page 154 in the algebra notebook

Think about complexification of real vector spaces.

page 154 in the algebra notebook

Let a,b:-|R. Let f,g : [a,b] -> |C. Let f,g be integrable.

Prove that

Integral([a,b],f+g) = Integral([a,b],f) + Integral([a,b],g).

Prove that

Integral([a,b],f+g) = Integral([a,b],f) + Integral([a,b],g).

Very easy.

Think about complexification of real vector spaces.

page 154 in the algebra notebook

Think about complexification of real vector spaces.

page 154 in the algebra notebook

Integral( 0, 4, 1/(1+sqrt(x)), dx ) = ???

4 - 2*log(3)

Integral( 0, 4, 1/(1+sqrt(x)), dx ) = 4 - 2*log(3)

hint: g(t)=t*t; later: log(1+t)

Integral( 0, 4, 1/(1+sqrt(x)), dx ) = 4 - 2*log(3)

hint: g(t)=t*t; later: log(1+t)

Integral( 0, log(2), sqrt( exp(x)-1 ), dx ) = ???

Integral( 0, log(2), sqrt( exp(x)-1 ), dx ) = 2 - pi/2

hint: g(t)=log(1+t*t); later: arctan

hint: g(t)=log(1+t*t); later: arctan