Math ASCII Notation Demo

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 f : ]0,oo[ x ]0,oo[ -> |R be the function satisfying:
/\(x>0,y>0) cos(arctan(y/x)) = f(x,y).
Find a formula for f(x,y) that does not involve trig functions.
f(x,y) = x / (x^2 + y^2)
page 95 in OLDTIMER
lim(x,y)->(0,0) (x^3 * y) / ((x^2 - y^2)^2 + y^2) = ???
lim(x,y)->(0,0) (x^3 * y) / ((x^2 - y^2)^2 + y^2) = 0
page 99 in OLDTIMER
lim(x,y)->(0,0) (x-sin(x))*y / ((x^2-y^2)^2 + y^2) = ???
lim(x,y)->(0,0) (x-sin(x))*y / ((x^2 - y^2)^2 + y^2) = 0
hint: x-sin(x) = x^3 * u(x), where lim(x->0) u(x) = 1/6.
further hint: |ab/(a^2+b^2)|<=K.
page 100 in OLDTIMER
Let Y : P(X) -> |R* satisfy:
(1) Y(O)=0.
(2) AcBcX => Y(A)<=Y(B).
(3) Y(AuB) <= Y(A)+Y(B).
Let M be a ring in X.
Let F = { EcX : /\a>0 \/A:-M Y(E+A) < a }.
In what sense can set F be said to be closed?
Define d:P(X)xP(X)->|R*, d(A,B)=Y(A+B).
Notice that d is a oo-pseudometric.
Now, if (1) /\n:-|N A[n]:-F and (2) lim(n->oo) d(A[n],A) = 0, then (3) A:-F.
See pages 21,64 in 2nd measure for relevant theorems.
Find the power series expansion of cos^2(z) about 0.
cos^2(z) = +(n=0 to oo) a[n]*z^(2n)
a[0] = 1
a[n] = (-1)^n * 2^(2n-1) / (2n)!
page 101 in OLDTIMER
Let 0 < R <= oo.
Let { c[n] }(n=0 to oo) be a sequence of complex numbers.
Let f : {z:-|C : |z|<R} -> |C be defined f(z) = +(n=0 to oo) c[n]*z^n.
Prove that f is infinitely diffable and derive the formula for its derivatives.
/\k:-|N /\|z|<R f|k|(z) = +(n=k to oo) c[n]* n!/(n-k)! * z^(n-k)
page 102 in OLDTIMER
/\k:-|N /\|z|<R f|k|(z) = +(n=0 to oo) c[n+k]* (n+k)!/(n!) * z^n
Let 0 < R <= oo.
Let { c[n] }(n=0 to oo) be a sequence of complex numbers.
Suppose that /\|z|<R +(n=0 to oo) c[n]*z^n = 0.
What can we conclude from this?
0 = c[0] = c[1] = c[2] = ... = c[n] = c[n+1] = ...
page 103 in OLDTIMER
Let 0 < R <= oo.
Let a[n],b[n] be sequences of complex numbers, indexed from zero.
Suppose that /\|z|<R +(n=0 to oo) a[n]*z^n = +(n=0 to oo) b[n]*z^n.
In the line above we understand that both series converge.
What can we conclude from this?
a[n] = b[n] for all n=0,1,2,3,4,...
page 104 in OLDTIMER
Prove that /\n:-|N 2^(2n) = +(k=0 to n) Newton(2n+1,2k).
page 106 in OLDTIMER
Interestingly, this formula can be derived by writing the power series expansion of cos(z)sin(z) in two different ways. One way involves the formula for multiplying series.
See page 104 in OLDTIMER.
Prove that /\n:-|N 2^(2n) = +(k=0 to n) Newton(2n+1,2k+1).
page 107 in OLDTIMER
Interestingly, this formula can be derived by writing the power series expansion of cos(z)sin(z) in two different ways. One way involves the formula for multiplying series.
See page 104 in OLDTIMER.