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.

Give the power series expansion of f(z) = z^2 * exp(z-1) about 1 and find its radius of convergence.
f(z) = +(n=0 to n=oo) a[n] * (z-1)^n
a[n] = 1/(n-2)! + 2/(n-1)! + 1/(n!)
hint: z^2 * exp(z-1) = ((z-1) + 1)^2 * exp(z-1)
f(z) = 6*sin(z^3) + z^3*(z^6 - 6)
Find the multiplicity of 0.
15
Determine the multiplicity of all zeros of cos.
1
hint: cos(z) = -sin(z - pi/2)
Determine the multiplicity of all zeros of sin.
1
page 87 in OLDTIMER
Give the power series expansion of cosine about pi/2.
+(k=0 to k=oo) [ (-1)^(k+1) * (z-pi/2)^(2k+1) / (2k+1)! ]
hint: cos(z) = -sin(z - pi/2)
Let G be an open connected subset of |C.
Let f,g : G -> |C be diffable.
What theorem can be used to prove that:
f*g=0 on G ==> f=0 on G or g=0 on G
Let G be an open connected subset of |C.
Let f : G -> |C be diffable.
(1) {z:-G : f(z)=0} has a limit point in G
(2) /\z:-G f(z)=0
thesis: (1)=>(2)
REMARK: MRW hasn't done this proof yet. 2001.02.23; 2001.05.15
Let W be a non-empty collection of sets such that
(1) A,B:-W => AnB:-W
(2) A,B:-W => A\B is a finite disjoint union of sets in W
Let y : W -> |R* be additive.
Can we conclude that y is finitely additive?
NO
W = { O, {1}, {2}, {3}, {1,2,3} }
y({1,2,3}) = 1
y(O) = y({1}) = y({2}) = y({3}) = 0
y is additive but not finitely additive
G c |C; f : G -> |C; a:-|C
What does it mean that function f has an isolated singularity at a?
(1) a doesn't belong to G
(2) there exists B = {z:-|C : 0 < |z-a| < r} c G such that f is diffable on B
G c |C; f : G -> |C; a:-|C
(1) a doesn't belong to G
(2) there exists B = {z:-|C : 0 < |z-a| < r} c G such that f is diffable on B
What is the verbal description of this situation?
function f has an isolated singularity at a
Let G be a subset of |C. Let f : G -> |C. Let a:-|C.
What does it mean that function f has a removable isolated singularity at a?
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
(4) g=f on B(a;r)\{a}