# 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.

/\n:-|N 2^(2n-1) = +( k=1 to k=n ) ???
2^(2n-1) = +(k=1 to n) Newton(2n,2k-1)
page 85 in OLDTIMER
Give the power series expansion of sin(z)sin(z) about 0.
sin(z)sin(z) = +(n=1 to n=oo) (-1)^(n+1) * 2^(2n-1) / (2n)! * z^(2n)
page 85 in OLDTIMER
hint1: 2^(2n-1), multiplication of series
hint2: 2sin(x)cos(x)=sin(2x), differentiation of power series
Integral( 0, pi, sin^4(x), dx ) = ???
You can use: Integral( 0, pi, sin2(x), dx ) = pi/2.
Integral( 0, pi, sin4(x), dx ) = 3/8 * pi
page 88 in OLDTIMER
Integral( 0, pi, cos^4(x), dx ) = ???
Integral( 0, pi, cos^4(x), dx ) = 3/8 * pi
page 89 in OLDTIMER
Let z be a complex number other than zero.
sin(Arg(z)) * |z| = ???
sin(Arg(z)) * |z| = Im(z)
hint: /\x:-|R sin(x) = Im(exp(ix))
page 90 in OLDTIMER
Let z be a complex number other than zero.
cos(Arg(z)) * |z| = ???
cos(Arg(z)) * |z| = Re(z)
hint: /\x:-|R cos(x) = Re(exp(ix))
page 90 in OLDTIMER
/\z:-|C\{0} Im(z) = |z| * sin( ??? )
/\z:-|C Im(z) = |z| * sin( Arg(z) )
page 90 in OLDTIMER
/\z:-|C\{0} Re(z) = |z| * cos( ??? )
/\z:-|C\{0} Re(z) = |z| * cos( Arg(z) )
page 90 in OLDTIMER
Let x>0, y>0, z = x + iy.
Prove that arctan(y/x) = Arg(z).
page 91 in OLDTIMER
D = {(x,y):-|C : x^2 + y^2 = y and x > 0}
Prove that /\z:-D |z| = sqrt(Im(z)) = ???
/\z:-D |z| = sqrt(Im(z)) = sin(Arg(z)) = sin(arctan(y/x))
page 91 in OLDTIMER