Complex.pm |
topic started 5/15/2002; 4:50:12 PM last post 5/15/2002; 4:50:12 PM |

Michael Gage - Complex.pm 5/15/2002; 4:50:12 PM (reads: 2201, responses: 0) |

## File path = /ww/webwork/pg/lib/Complex.pm
- NAME
- SYNOPSIS
- DESCRIPTION
- OPERATIONS
- CREATION
- STRINGIFICATION
- USAGE
- ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
- ERRORS DUE TO INDIGESTIBLE ARGUMENTS
- BUGS
- AUTHORS
## NAMEMath::Complex - complex numbers and associated mathematical functions
## SYNOPSISuse Math::Complex; $z = Math::Complex->make(5, 6);
## DESCRIPTIONThis package lets you create and manipulate complex numbers. By default,
If you wonder what complex numbers are, they were invented to be able to solve the following equation: x*x = -1 and by definition, the solution is noted The arithmetics with pure imaginary numbers works just like you would expect it with real numbers... you just have to remember that i*i = -1 so you have: 5i + 7i = i * (5 + 7) = 12i Complex numbers are numbers that have both a real part and an imaginary part, and are usually noted: a + bi where (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i A graphical representation of complex numbers is possible in a plane
(also called the z = a + bi is the point whose coordinates are (a, b). Actually, it would be the vector originating from (0, 0) to (a, b). It follows that the addition of two complex numbers is a vectorial addition. Since there is a bijection between a point in the 2D plane and a complex number (i.e. the mapping is unique and reciprocal), a complex number can also be uniquely identified with polar coordinates: [rho, theta] where rho * exp(i * theta) where a = rho * cos(theta) which is also expressed by this formula: z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) In other words, it's the projection of the vector onto the The polar notation (also known as the trigonometric
representation) is much more handy for performing multiplications and
divisions of complex numbers, whilst the cartesian notation is better
suited for additions and subtractions. Real numbers are on the All the common operations that can be performed on a real number have
been defined to work on complex numbers as well, and are merely
For instance, the sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i It can also be extended to be an application from sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) Indeed, a negative real number can be noted sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i which is exactly what we had defined for negative real numbers above.
The All the common mathematical functions defined on real numbers that
are extended to complex numbers share that same property of working
A z = a + bi Simple... Now look: z * ~z = (a + bi) * (a - bi) = a*a + b*b We saw that the norm of rho = abs(z) = sqrt(a*a + b*b) so z * ~z = abs(z) ** 2 If z is a pure real number (i.e. a * a = abs(a) ** 2 which is true (
## OPERATIONSGiven the following notations: z1 = a + bi = r1 * exp(i * t1) the following (overloaded) operations are supported on complex numbers: z1 + z2 = (a + c) + i(b + d) The following extra operations are supported on both real and complex numbers: Re(z) = a cbrt(z) = z ** (1/3) tan(z) = sin(z) / cos(z) csc(z) = 1 / sin(z) asin(z) = -i * log(i*z + sqrt(1-z*z)) acsc(z) = asin(1 / z) sinh(z) = 1/2 (exp(z) - exp(-z)) csch(z) = 1 / sinh(z) asinh(z) = log(z + sqrt(z*z+1)) acsch(z) = asinh(1 / z)
The 1 + j + j*j = 0; is a simple matter of writing: $j = ((root(1, 3))[1]; The (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) The
## CREATIONTo create a complex number, use either: $z = Math::Complex->make(3, 4); if you know the cartesian form of the number, or $z = 3 + 4*i; if you like. To create a number using the polar form, use either: $z = Math::Complex->emake(5, pi/3); instead. The first argument is the modulus, the second is the angle
(in radians, the full circle is 2*pi). (Mnemonic: It is possible to write: $x = cplxe(-3, pi/4); but that will be silently converted into It is also possible to have a complex number as either argument of
either the $z1 = cplx(-2, 1);
## STRINGIFICATIONWhen printed, a complex number is usually shown under its cartesian
form By calling the routine This default can be overridden on a per-number basis by calling the
For instance: use Math::Complex; Complex1::display_format('polar'); The polar format attempts to emphasize arguments like
## USAGEThanks to overloading, the handling of arithmetics with complex numbers is simple and almost transparent. Here are some examples: use Math::Complex; $j = cplxe(1, 2*pi/3); # $j ** 3 == 1 $z = -16 + 0*i; # Force it to be a complex $k = exp(i * 2*pi/3); $z->Re(3); # Re, Im, arg, abs,
## ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZEROThe division (/) and the following functions log ln log10 logn cannot be computed for all arguments because that would mean dividing by zero or taking logarithm of zero. These situations cause fatal runtime errors looking like this cot(0): Division by zero. or atanh(-1): Logarithm of zero. For the Note that because we are operating on approximations of real numbers,
these errors can happen when merely `too close' to the singularities
listed above. For example
## ERRORS DUE TO INDIGESTIBLE ARGUMENTSThe Complex1::make: Cannot take real part of ...
## BUGSSaying All routines expect to be given real or complex numbers. Don't attempt to use BigFloat, since Perl has currently no rule to disambiguate a '+' operation (for instance) between two overloaded entities. In Cray UNICOS there is some strange numerical instability that results in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware. The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex. Whatever it is, it does not manifest itself anywhere else where Perl runs.
## AUTHORSRaphael Manfredi < Extensive patches by Daniel S. Lewart < ## File path = /ww/webwork/pg/lib/Complex1.pm |