Math::Complex(3) Perl Programmers Reference Guide Math::Complex(3) NAME Math::Complex - complex numbers and associated mathematical functions SYNOPSIS use Math::Complex; $z = Math::Complex->make(5, 6); $t = 4 - 3*i + $z; $j = cplxe(1, 2*pi/3); DESCRIPTION This package lets you create and manipulate complex numbers. By default,Perllimits itself to real numbers, but an extra use statement brings full complex support, along with a full set of mathematical functions typically associated with and/or extended to complex numbers. 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 notedi(engineers usejinstead sinceiusually denotes an intensity, but the name does not matter). The numberiis a pureimaginarynumber. 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 4i - 3i = i * (4 - 3) = i 4i * 2i = -8 6i / 2i = 3 1 / i = -i Complex numbers are numbers that have both a real part and an imaginary part, and are usually noted: a + bi where a is therealpart and b is theimaginarypart. The arithmetic with complex numbers is straightforward. You have to keep track of the real and the imaginary parts, but otherwise the rules used for real numbers just apply: (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i A graphical representation of complex numbers is possible in a plane (also called thecomplex plane, but it's really a 2D plane). The number 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 is the distance to the origin, and theta the angle between the vector and thexaxis. There is a notation for this using the exponential form, which is: rho * exp(i * theta) whereiis the famous imaginary number introduced above. Conversion between this form and the cartesian form a + bi is immediate: a = rho * cos(theta) b = rho * sin(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 thexandyaxes. Mathematicians callrhothenormormodulusandthetatheargumentof the complex number. Thenormof z will be noted abs(z). 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 thexaxis, and thereforethetais zero orpi. All the common operations that can be performed on a real number have been defined to work on complex numbers as well, and are merelyextensionsof the operations defined on real numbers. This means they keep their natural meaning when there is no imaginary part, provided the number is within their definition set. For instance, the sqrt routine which computes the square root of its argument is only defined for non-negative real numbers and yields a non-negative real number (it is an application fromR+toR+). If we allow it to return a complex number, then it can be extended to negative real numbers to become an application fromRtoC(the set of complex numbers): sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i It can also be extended to be an application fromCtoC, whilst its restriction toRbehaves as defined above by using the following definition: sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) Indeed, a negative real number can be noted [x,pi] (the modulusxis always non-negative, so [x,pi] is really -x, a negative number) and the above definition states that 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 sqrt returns only one of the solutions: if you want the both, use the root function. All the common mathematical functions defined on real numbers that are extended to complex numbers share that same property of workingasusualwhen the imaginary part is zero (otherwise, it would not be called an extension, would it?). Anewoperation possible on a complex number that is the identity for real numbers is called theconjugate, and is noted with an horizontal bar above the number, or ~z here. z = a + bi ~z = a - bi Simple... Now look: z * ~z = (a + bi) * (a - bi) = a*a + b*b We saw that the norm of z was noted abs(z) and was defined as the distance to the origin, also known as: rho = abs(z) = sqrt(a*a + b*b) so z * ~z = abs(z) ** 2 If z is a pure real number (i.e. b == 0), then the above yields: a * a = abs(a) ** 2 which is true (abs has the regular meaning for real number, i.e. stands for the absolute value). This example explains why the norm of z is noted abs(z): it extends the abs function to complex numbers, yet is the regular abs we know when the complex number actually has no imaginary part... This justifiesa posterioriour use of the abs notation for the norm. OPERATIONS Given the following notations: z1 = a + bi = r1 * exp(i * t1) z2 = c + di = r2 * exp(i * t2) z = <any complex or real number> the following (overloaded) operations are supported on complex numbers: z1 + z2 = (a + c) + i(b + d) z1 - z2 = (a - c) + i(b - d) z1 * z2 = (r1 * r2) * exp(i * (t1 + t2)) z1 / z2 = (r1 / r2) * exp(i * (t1 - t2)) z1 ** z2 = exp(z2 * log z1) ~z = a - bi abs(z) = r1 = sqrt(a*a + b*b) sqrt(z) = sqrt(r1) * exp(i * t/2) exp(z) = exp(a) * exp(i * b) log(z) = log(r1) + i*t sin(z) = 1/2i (exp(i * z1) - exp(-i * z)) cos(z) = 1/2 (exp(i * z1) + exp(-i * z)) atan2(z1, z2) = atan(z1/z2) The following extra operations are supported on both real and complex numbers: Re(z) = a Im(z) = b arg(z) = t abs(z) = r cbrt(z) = z ** (1/3) log10(z) = log(z) / log(10) logn(z, n) = log(z) / log(n) tan(z) = sin(z) / cos(z) csc(z) = 1 / sin(z) sec(z) = 1 / cos(z) cot(z) = 1 / tan(z) asin(z) = -i * log(i*z + sqrt(1-z*z)) acos(z) = -i * log(z + i*sqrt(1-z*z)) atan(z) = i/2 * log((i+z) / (i-z)) acsc(z) = asin(1 / z) asec(z) = acos(1 / z) acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) sinh(z) = 1/2 (exp(z) - exp(-z)) cosh(z) = 1/2 (exp(z) + exp(-z)) tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z)) csch(z) = 1 / sinh(z) sech(z) = 1 / cosh(z) coth(z) = 1 / tanh(z) asinh(z) = log(z + sqrt(z*z+1)) acosh(z) = log(z + sqrt(z*z-1)) atanh(z) = 1/2 * log((1+z) / (1-z)) acsch(z) = asinh(1 / z) asech(z) = acosh(1 / z) acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))arg,abs,log,csc,cot,acsc,acot,csch,coth,acosech,acotanh, have aliasesrho,theta,ln,cosec,cotan,acosec,acotan,cosech,cotanh,acosech,acotanh, respectively. Re, Im, arg, abs, rho, and theta can be used also also mutators. The cbrt returns only one of the solutions: if you want all three, use the root function. Therootfunction is available to compute all thenroots of some complex, wherenis a strictly positive integer. There are exactlynsuch roots, returned as a list. Getting the number mathematicians call j such that: 1 + j + j*j = 0; is a simple matter of writing: $j = ((root(1, 3))[1]; Thekth root for z = [r,t] is given by: (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) Thespaceshipcomparison operator, <=>, is also defined. In order to ensure its restriction to real numbers is conform to what you would expect, the comparison is run on the real part of the complex number first, and imaginary parts are compared only when the real parts match. CREATION To create a complex number, use either: $z = Math::Complex->make(3, 4); $z = cplx(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); $x = cplxe(5, pi/3); instead. The first argument is the modulus, the second is the angle (in radians, the full circle is 2*pi). (Mnemonic: e is used as a notation for complex numbers in the polar form). It is possible to write: $x = cplxe(-3, pi/4); but that will be silently converted into [3,-3pi/4], since the modulus must be non-negative (it represents the distance to the origin in the complex plane). It is also possible to have a complex number as either argument of either the make or emake: the appropriate component of the argument will be used. $z1 = cplx(-2, 1); $z2 = cplx($z1, 4); STRINGIFICATION When printed, a complex number is usually shown under its cartesian forma+bi, but there are legitimate cases where the polar format[r,t]is more appropriate. By calling the routine Math::Complex::display_format and supplying either "polar" or "cartesian", you override the default display format, which is "cartesian". Not supplying any argument returns the current setting. This default can be overridden on a per-number basis by calling the display_format method instead. As before, not supplying any argument returns the current display format for this number. Otherwise whatever you specify will be the new display format forthisparticular number. For instance: use Math::Complex; Math::Complex::display_format('polar'); $j = ((root(1, 3))[1]; print "j = $j\n"; # Prints "j = [1,2pi/3] $j->display_format('cartesian'); print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i" The polar format attempts to emphasize arguments likek*pi/n(wherenis a positive integer andkan integer within [-9,+9]). USAGE Thanks 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 print "j = $j, j**3 = ", $j ** 3, "\n"; print "1 + j + j**2 = ", 1 + $j + $j**2, "\n"; $z = -16 + 0*i; # Force it to be a complex print "sqrt($z) = ", sqrt($z), "\n"; $k = exp(i * 2*pi/3); print "$j - $k = ", $j - $k, "\n"; $z->Re(3); # Re, Im, arg, abs, $j->arg(2); # (the last two aka rho, theta) # can be used also as mutators. ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO The division (/) and the following functions log ln log10 logn tan sec csc cot atan asec acsc acot tanh sech csch coth atanh asech acsch acoth 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. (Because in the definition of cot(0), the divisor sin(0) is 0) Died at ... or atanh(-1): Logarithm of zero. Died at... For the csc, cot, asec, acsc, acot, csch, coth, asech, acsch, the argument cannot be 0 (zero). For the the logarithmic functions and the atanh, acoth, the argument cannot be 1 (one). For the atanh, acoth, the argument cannot be -1 (minus one). For the atan, acot, the argument cannot be i (the imaginary unit). For the atan, acoth, the argument cannot be -i (the negative imaginary unit). For the tan, sec, tanh, the argument cannot bepi/2 + k * pi, wherekis any integer. 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 tan(2*atan2(1,1)+1e-15) will die of division by zero. ERRORS DUE TO INDIGESTIBLE ARGUMENTS The make and emake accept both real and complex arguments. When they cannot recognize the arguments they will die with error messages like the following Math::Complex::make: Cannot take real part of ... Math::Complex::make: Cannot take real part of ... Math::Complex::emake: Cannot take rho of ... Math::Complex::emake: Cannot take theta of ... BUGS Saying use Math::Complex; exports many mathematical routines in the caller environment and even overrides some (sqrt, log). This is construed as a feature by the Authors, actually... ;-) 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 inroot(),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. AUTHORS Raphael Manfredi <Raphael_Manfredi@grenoble.hp.com> and Jarkko Hietaniemi <jhi@iki.fi>. Extensive patches by Daniel S. Lewart <d-lewart@uiuc.edu>. 3rd Berkeley Distribution perl 5.005, patch 02 Math::Complex(3)