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

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

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.

You can use: Integral( 0, pi, sin2(x), dx ) = pi/2.

Integral( 0, pi, sin4(x), dx ) = 3/8 * pi

page 88 in OLDTIMER

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

page 89 in OLDTIMER

Let z be a complex number other than zero.

sin(Arg(z)) * |z| = ???

sin(Arg(z)) * |z| = ???

sin(Arg(z)) * |z| = Im(z)

hint: /\x:-|R sin(x) = Im(exp(ix))

page 90 in OLDTIMER

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| = ???

cos(Arg(z)) * |z| = Re(z)

hint: /\x:-|R cos(x) = Re(exp(ix))

page 90 in OLDTIMER

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

page 90 in OLDTIMER

/\z:-|C\{0} Re(z) = |z| * cos( ??? )

/\z:-|C\{0} Re(z) = |z| * cos( Arg(z) )

page 90 in OLDTIMER

page 90 in OLDTIMER

Let x>0, y>0, z = x + iy.

Prove that arctan(y/x) = Arg(z).

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)) = ???

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

page 91 in OLDTIMER