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 z be a complex number other than zero. Let n be a natural
number. Describe the set {w:-C : w^n = z}.

|z|^(1/n) * exp( (i*(arg(z) + k*2*pi)) / n )

for some k=0,1,...,n-1

for some k=0,1,...,n-1

Let p be a real number.

What is the radius of convergence of the power series (n^p * z^n)?

What is the radius of convergence of the power series (n^p * z^n)?

R = 1

Find the radius of convergence of the power series

z^(2*n) / (n * 2^n).

z^(2*n) / (n * 2^n).

the square root of 2

Let z be a complex number such that |z| >= 1.

Prove that the series (z^n) diverges.

Prove that the series (z^n) diverges.

(1) |z|>=1

(2) not (|z|^n -> 0)

(3) not (|z^n| -> 0)

(4) not ( z^n -> 0)

(5) series (z^n) diverges

page 189 in OLDTIMER

(2) not (|z|^n -> 0)

(3) not (|z^n| -> 0)

(4) not ( z^n -> 0)

(5) series (z^n) diverges

page 189 in OLDTIMER

(a+b)! / (a! * b!) = ???

the Newton binomial

1) a+b over a

2) a+b over b

1) a+b over a

2) a+b over b

Define sin(z) for complex numbers using a series.

sin(z) = +(n=0 to oo) [ (-1)^n * z^(2n+1) / (2n+1)! ]

page 72 in palace

page 72 in palace

Define cos(z) for complex numbers using a series.

cos(z) = +(n=0 to oo) [ (-1)^n * z^(2n) / (2n)! ]

page 72 in palace

page 72 in palace

Define cos(z) for complex numbers using exp.

cos(z) = ( exp(iz) + exp(-iz) ) / 2

page 69 in palace

page 69 in palace

Define sin(z) for complex numbers using exp.

sin(z) = ( exp(iz) - exp(-iz) ) / 2i

page 69 in palace

page 69 in palace

Given that exp(x), cos(x), sin(x) are defined for real numbers,
define exp(z) for complex numbers.

exp(x + iy) = exp(x) * ( cos(y) + i*sin(y) )