block({
  include(`complex-prg.txt'):
  let(complexpi,4*atan(1)):

  func(realoddroot,{sgn([x])*(abs([x])^(1/[y]))}):

  print_`Real powers and complex functions\n\n':

  print_`1. Non-integer powers with positive real bases\n':
  print_`9^0.5        = ' (9^0.5) `\n':
  print_`2^1.5        = ' (2^1.5) `\n':
  print_`27^(1/3)     = ' (27^(1/3)) `\n':
  print_`These calculations have real results and use ordinary floating point.\n\n':

  print_`2. Complex representation and arithmetic\n':
  print_`<a,b> represents a + b*i; I from the module is <0,1>.\n':
  let(z,<3,4>):
  let(w,<1,-2>):
  print_`z            = ' [z] `\n':
  print_`w            = ' [w] `\n':
  print_`z + w        = ' cadd([z],[w]) `\n':
  print_`z - w        = ' csub([z],[w]) `\n':
  print_`z * w        = ' cmul([z],[w]) `\n':
  print_`z / w        = ' cdiv([z],[w]) `\n':
  print_`conjugate(z) = ' conj([z]) `\n':
  print_`abs(z)       = ' cmag([z]) `\n':
  print_`i*i          = ' cmul([I],[I]) `\n':
  print_`Vector *, / and ^ act componentwise. Use cmul, cdiv and cpow for complex arithmetic.\n\n':

  print_`3. Polar coordinates\n':
  print_`cpol(z) returns <magnitude,angle>; angles are in radians.\n':
  print_`cpol(<3,4>)  = ' cpol([z]) `\n':
  print_`pcart(cpol(z)) = ' pcart(cpol([z])) `\n\n':

  print_`4. Fractional powers of negative real numbers\n':
  print_`Scalar (-4)^0.5 raises an error; it does not promote to complex arithmetic.\n':
  print_`cpow(<-4,0>,0.5) = ' cpow(<-4,0>,0.5) `\n':
  print_`csqrt(<-1,0>)    = ' csqrt(<-1,0>) `\n':
  print_`cpow(z,p) uses a real exponent and the principal angle of z.\n':
  print_`Principal cube root of -8 = ' cpow(<-8,0>,1/3) `\n':
  print_`Real cube root of -8      = ' quote(realoddroot(-8,3)) `\n':
  print_`realoddroot(x,n) assumes a positive odd integer n.\n':
  print_`A negative real number can have a real odd root as well as other complex roots.\n\n':

  print_`5. Enumerating roots\n':
  print_`csqrt, croot and croots are provided by complex-prg.txt.\n':
  print_`croot(z,n,k) adds 2*pi*k to the angle before dividing by n.\n':
  print_`Use positive integer n and branch k from 0 to n-1.\n':
  print_`All cube roots of -8 = ' croots(<-8,0>,3) `\n':
  print_`All fourth roots of 1 = ' croots(<1,0>,4) `\n\n':

  print_`6. Complex exponents and logarithms\n':
  print_`cpow2(z,w) accepts a complex exponent w and returns a principal value.\n':
  print_`i^i              = ' cpow2([I],[I]) `\n':
  print_`exp(-pi/2)       = ' eul(-[complexpi]/2) `\n':
  print_`(2+3i)^(0.5-0.25i) = ' cpow2(<2,3>,<0.5,-0.25>) `\n':
  print_`clog(z)          = ' clog([z]) `\n':
  print_`clog(<-1,0>)     = ' clog(<-1,0>) `\n':
  print_`Logarithms and general powers in these examples use nonzero bases.\n\n':

  print_`7. Complex exponential and trigonometric functions\n':
  print_`cexp, csin, ccos and ctan are provided by complex-prg.txt.\n':
  print_`cexp(<a,b>) = exp(a)*<cos(b),sin(b)>\n':
  print_`exp(i*pi)        = ' cexp(<0,[complexpi]>) `\n':
  print_`exp(log(z))      = ' cexp(clog([z])) `\n':
  print_`sin(i)           = ' csin([I]) `\n':
  print_`cos(i)           = ' ccos([I]) `\n':
  print_`tan(i)           = ' ctan([I]) `\n':
  print_`cos(z)^2+sin(z)^2 = ' cadd(cmul(ccos([z]),ccos([z])),cmul(csin([z]),csin([z]))) `\n\n':

  print_`8. Differentiating a complex function\n':
  print_`cderiv(function,point,step) uses a centered finite difference.\n':
  print_`d(exp)/dz at i   = ' cderiv(cexp,[I],<0.00001,0>) `\n':
  print_`exp(i)           = ' cexp([I])
})