```{index} single: Math::Complex ``` # Math::Complex complex numbers and associated mathematical functions ## SYNOPSIS ```perl 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, _Perl_ limits 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: ```text x*x = -1 ``` and by definition, the solution is noted _i_ (engineers use _j_ instead since _i_ usually denotes an intensity, but the name does not matter). The number _i_ is a pure _imaginary_ number. The arithmetics with pure imaginary numbers works just like you would expect it with real numbers... you just have to remember that ```text i*i = -1 ``` so you have: ```text 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: ```text a + bi ``` where `a` is the _real_ part and `b` is the _imaginary_ part. 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: ```text (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 the _complex plane_, but it's really a 2D plane). The number ```text 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: ```text [rho, theta] ``` where `rho` is the distance to the origin, and `theta` the angle between the vector and the _x_ axis. There is a notation for this using the exponential form, which is: ```text rho * exp(i * theta) ``` where _i_ is the famous imaginary number introduced above. Conversion between this form and the cartesian form `a + bi` is immediate: ```text a = rho * cos(theta) b = rho * sin(theta) ``` which is also expressed by this formula: ```text z = rho * exp(i * theta) = rho * (cos theta + i * sin theta) ``` In other words, it's the projection of the vector onto the _x_ and _y_ axes. Mathematicians call _rho_ the _norm_ or _modulus_ and _theta_ the _argument_ of the complex number. The _norm_ of `z` is marked here as `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 the _x_ axis, and therefore _y_ or _theta_ is zero or _pi_. 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 _extensions_ of 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 from **R+** to **R+**). If we allow it to return a complex number, then it can be extended to negative real numbers to become an application from **R** to **C** (the set of complex numbers): ```text sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i ``` It can also be extended to be an application from **C** to **C**, whilst its restriction to **R** behaves as defined above by using the following definition: ```text sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2) ``` Indeed, a negative real number can be noted `[x,pi]` (the modulus _x_ is always non-negative, so `[x,pi]` is really `-x`, a negative number) and the above definition states that ```text 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 working _as usual_ when the imaginary part is zero (otherwise, it would not be called an extension, would it?). A _new_ operation possible on a complex number that is the identity for real numbers is called the _conjugate_, and is noted with a horizontal bar above the number, or `~z` here. ```text z = a + bi ~z = a - bi ``` Simple... Now look: ```text 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: ```text rho = abs(z) = sqrt(a*a + b*b) ``` so ```text z * ~z = abs(z) ** 2 ``` If z is a pure real number (i.e. `b == 0`), then the above yields: ```text 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 justifies _a posteriori_ our use of the `abs` notation for the norm. ## OPERATIONS Given the following notations: ```text z1 = a + bi = r1 * exp(i * t1) z2 = c + di = r2 * exp(i * t2) z = ``` the following (overloaded) operations are supported on complex numbers: ```text 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(y, x) = atan(y / x) # Minding the right quadrant, note the order. ``` The definition used for complex arguments of atan2() is ```text -i log((x + iy)/sqrt(x*x+y*y)) ``` Note that atan2(0, 0) is not well-defined. The following extra operations are supported on both real and complex numbers: ```text 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 aliases _rho_, _theta_, _ln_, _cosec_, _cotan_, _acosec_, _acotan_, _cosech_, _cotanh_, _acosech_, _acotanh_, respectively. `Re`, `Im`, `arg`, `abs`, `rho`, and `theta` can be used also as mutators. The `cbrt` returns only one of the solutions: if you want all three, use the `root` function. The _root_ function is available to compute all the _n_ roots of some complex, where _n_ is a strictly positive integer. There are exactly _n_ such roots, returned as a list. Getting the number mathematicians call `j` such that: ```perl 1 + j + j*j = 0; ``` is a simple matter of writing: ```perl $j = (root(1, 3))[1]; ``` The _k_th root for `z = [r,t]` is given by: ```text (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n) ``` You can return the _k_th root directly by `root(z, n, k)`, indexing starting from _zero_ and ending at _n - 1_. The _spaceship_ numeric comparison 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: ```perl $z = Math::Complex->make(3, 4); $z = cplx(3, 4); ``` if you know the cartesian form of the number, or ```perl $z = 3 + 4*i; ``` if you like. To create a number using the polar form, use either: ```perl $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: ```perl $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 the `make`, `emake`, `cplx`, and `cplxe`: the appropriate component of the argument will be used. ```perl $z1 = cplx(-2, 1); $z2 = cplx($z1, 4); ``` The `new`, `make`, `emake`, `cplx`, and `cplxe` will also understand a single (string) argument of the forms ```text 2-3i -3i [2,3] [2,-3pi/4] [2] ``` in which case the appropriate cartesian and exponential components will be parsed from the string and used to create new complex numbers. The imaginary component and the theta, respectively, will default to zero. The `new`, `make`, `emake`, `cplx`, and `cplxe` will also understand the case of no arguments: this means plain zero or (0, 0). ## DISPLAYING When printed, a complex number is usually shown under its cartesian style _a+bi_, but there are legitimate cases where the polar style _\[r,t\]_ is more appropriate. The process of converting the complex number into a string that can be displayed is known as _stringification_. By calling the class method `Math::Complex::display_format` and supplying either `"polar"` or `"cartesian"` as an argument, you override the default display style, which is `"cartesian"`. Not supplying any argument returns the current settings. 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 style for this number. Otherwise whatever you specify will be the new display style for _this_ particular number. For instance: ```perl 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 style attempts to emphasize arguments like _k\*pi/n_ (where _n_ is a positive integer and _k_ an integer within \[-9, +9\]), this is called _polar pretty-printing_. For the reverse of stringifying, see the `make` and `emake`. ### CHANGED IN PERL 5.6 The `display_format` class method and the corresponding `display_format` object method can now be called using a parameter hash instead of just a one parameter. The old display format style, which can have values `"cartesian"` or `"polar"`, can be changed using the `"style"` parameter. ```perl $j->display_format(style => "polar"); ``` The one parameter calling convention also still works. ```perl $j->display_format("polar"); ``` There are two new display parameters. The first one is `"format"`, which is a sprintf()-style format string to be used for both numeric parts of the complex number(s). The is somewhat system-dependent but most often it corresponds to `"%.15g"`. You can revert to the default by setting the `format` to `undef`. ```perl ## the $j from the above example $j->display_format('format' => '%.5f'); print "j = $j\n"; # Prints "j = -0.50000+0.86603i" $j->display_format('format' => undef); print "j = $j\n"; # Prints "j = -0.5+0.86603i" ``` Notice that this affects also the return values of the `display_format` methods: in list context the whole parameter hash will be returned, as opposed to only the style parameter value. This is a potential incompatibility with earlier versions if you have been calling the `display_format` method in list context. The second new display parameter is `"polar_pretty_print"`, which can be set to true or false, the default being true. See the previous section for what this means. ## USAGE Thanks to overloading, the handling of arithmetics with complex numbers is simple and almost transparent. Here are some examples: ```perl 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. ``` ## CONSTANTS ### PI The constant `pi` and some handy multiples of it (pi2, pi4, and pip2 (pi/2) and pip4 (pi/4)) are also available if separately exported: ```perl use Math::Complex ':pi'; $third_of_circle = pi2 / 3; ``` ### Inf The floating point infinity can be exported as a subroutine Inf(): ```perl use Math::Complex qw(Inf sinh); my $AlsoInf = Inf() + 42; my $AnotherInf = sinh(1e42); print "$AlsoInf is $AnotherInf\n" if $AlsoInf == $AnotherInf; ``` Note that the stringified form of infinity varies between platforms: it can be for example any of ```text inf infinity INF 1.#INF ``` or it can be something else. Also note that in some platforms trying to use the infinity in arithmetic operations may result in Perl crashing because using an infinity causes SIGFPE or its moral equivalent to be sent. The way to ignore this is ```perl local $SIG{FPE} = sub { }; ``` ## ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO The division (/) and the following functions ```text 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 ```text cot(0): Division by zero. (Because in the definition of cot(0), the divisor sin(0) is 0) Died at ... ``` or ```text 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 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 be _pi/2 + k \* pi_, where _k_ is any integer. atan2(0, 0) is undefined, and if the complex arguments are used for atan2(), a division by zero will happen if z1\*\*2+z2\*\*2 == 0. Note that because we are operating on approximations of real numbers, these errors can happen when merely \`too close' to the singularities listed above. ## 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 ```text 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`, `atan2`). 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 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. ## SEE ALSO [Math::Trig](Trig.md) ## AUTHORS Daniel S. Lewart <`lewart!at!uiuc.edu`>, Jarkko Hietaniemi <`jhi!at!iki.fi`>, Raphael Manfredi <`Raphael_Manfredi!at!pobox.com`>, Zefram ## LICENSE This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself.