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 (X,dX), (Y,dY) be metric spaces. Let f : XxY -> |R.

(1) /\x0:-X /\y0:-Y /\E>0 \/A>0 /\y:-Y [ dY(y,y0)<A ==> |f(x0,y)-f(x0,y0)|<E ]

(2) /\x0:-X /\E>0 \/A>0 /\x:-X /\y:-Y [ dX(x,x0)<A ==> |f(x,y)- f(x0,y)|<E ]

What can we say about function f?

(1) /\x0:-X /\y0:-Y /\E>0 \/A>0 /\y:-Y [ dY(y,y0)<A ==> |f(x0,y)-f(x0,y0)|<E ]

(2) /\x0:-X /\E>0 \/A>0 /\x:-X /\y:-Y [ dX(x,x0)<A ==> |f(x,y)- f(x0,y)|<E ]

What can we say about function f?

It is continuous. The proof is swift.

Let z,w be complex numbers, z != 0.

exp(w) = z <=> ???

exp(w) = z <=> ???

Re(w)=log|z| and Im(w)=Arg(z)+k*2*PI, for some integer k

w = log|z| + i*Arg(z) + k+2*PI*i, for some integer k

page 60,36 in OLDTIMER

w = log|z| + i*Arg(z) + k+2*PI*i, for some integer k

page 60,36 in OLDTIMER

Arg : |C \ {0} -> ]-PI,PI] ; z = |z|*exp(i*Arg(z)).

Where is Arg continuous?

Where is Arg continuous?

|C \ {z<=0}

page 38 in OLDTIMER

page 38 in OLDTIMER

Let D be an open subset of |R^2.

Let f : D -> |R.

Suppose that f has bounded partial derivatives.

Does f have to be continuous?

Let f : D -> |R.

Suppose that f has bounded partial derivatives.

Does f have to be continuous?

Yes.

page 40 in OLDTIMER

page 40 in OLDTIMER

Let D be an open subset of |R^2.

Let f : D -> |R. Suppose that the partial derivative fy exists on D.

(1) /\(x0,y0):-D /\E>0 \/A>0 /\(x,y0):-D |x-x0|<A ==> |f(x,y0)-f(x0,y0)|<E

(2) \/M>0 /\(x,y):-D |fy(x,y)|<M

Let f : D -> |R. Suppose that the partial derivative fy exists on D.

(1) /\(x0,y0):-D /\E>0 \/A>0 /\(x,y0):-D |x-x0|<A ==> |f(x,y0)-f(x0,y0)|<E

(2) \/M>0 /\(x,y):-D |fy(x,y)|<M

Yes.

page 41 in OLDTIMER

page 41 in OLDTIMER

Let D,E be open subsets of |C.

f : D -> |C; g : E -> |C; f(D) c E

(1) f is continuous on D

(2) g is diffable on f(D)

(3) /\z:-D g'(f(z)) != 0

(4) /\z:-D g(f(z))=z

f : D -> |C; g : E -> |C; f(D) c E

(1) f is continuous on D

(2) g is diffable on f(D)

(3) /\z:-D g'(f(z)) != 0

(4) /\z:-D g(f(z))=z

It is diffable and /\z:-D f'(z)=1/g'(f(z)).

page 42 in OLDTIMER

page 42 in OLDTIMER

D = |C \ {z<=0}. f : D -> |C. k is an integer.

f(z) = log|z| + i*(Arg(z) + 2*k*PI)

Is f diffable?

f(z) = log|z| + i*(Arg(z) + 2*k*PI)

Is f diffable?

Yes.

/\z:-D f'(z) = 1/z

page 43 in OLDTIMER

/\z:-D f'(z) = 1/z

page 43 in OLDTIMER

E = |C \ {z>=0}. Let w : E -> |C.

(1) /\z:-E exp(w(z)) = z

(2) w is continuous

Does such a function exist?

(1) /\z:-E exp(w(z)) = z

(2) w is continuous

Does such a function exist?

Yes.

If Im(z)>=0, then w(z) = log|z| + i*Arg(z).

If Im(z)<0, then w(z) = log|z| + i*Arg(z) + i*2*PI.

page 43 in OLDTIMER

If Im(z)>=0, then w(z) = log|z| + i*Arg(z).

If Im(z)<0, then w(z) = log|z| + i*Arg(z) + i*2*PI.

page 43 in OLDTIMER

E = |C \ {0}. Let w : E -> |C.

(1) /\z:-E exp(w(z)) = z

(2) w is continuous

Does such a function exist?

(1) /\z:-E exp(w(z)) = z

(2) w is continuous

Does such a function exist?

No.

page 45 in OLDTIMER

page 45 in OLDTIMER

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

Suppose that h is diffable.

What can we say about function Re(h)?

Suppose that h is diffable.

What can we say about function Re(h)?

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

page 46 in OLDTIMER

page 46 in OLDTIMER