{
    "mode": "perldoc",
    "parameter": "Math::Complex",
    "section": "",
    "url": "https://www.chedong.com/phpMan.php/perldoc/Math%3A%3AComplex/json",
    "generated": "2026-06-13T21:21:11Z",
    "synopsis": "use Math::Complex;\n$z = Math::Complex->make(5, 6);\n$t = 4 - 3*i + $z;\n$j = cplxe(1, 2*pi/3);",
    "sections": {
        "NAME": {
            "content": "Math::Complex - complex numbers and associated mathematical functions\n",
            "subsections": []
        },
        "SYNOPSIS": {
            "content": "use Math::Complex;\n\n$z = Math::Complex->make(5, 6);\n$t = 4 - 3*i + $z;\n$j = cplxe(1, 2*pi/3);\n",
            "subsections": []
        },
        "DESCRIPTION": {
            "content": "This package lets you create and manipulate complex numbers. By default, *Perl* limits itself to\nreal numbers, but an extra \"use\" statement brings full complex support, along with a full set of\nmathematical functions typically associated with and/or extended to complex numbers.\n\nIf you wonder what complex numbers are, they were invented to be able to solve the following\nequation:\n\nx*x = -1\n\nand by definition, the solution is noted *i* (engineers use *j* instead since *i* usually\ndenotes an intensity, but the name does not matter). The number *i* is a pure *imaginary*\nnumber.\n\nThe arithmetics with pure imaginary numbers works just like you would expect it with real\nnumbers... you just have to remember that\n\ni*i = -1\n\nso you have:\n\n5i + 7i = i * (5 + 7) = 12i\n4i - 3i = i * (4 - 3) = i\n4i * 2i = -8\n6i / 2i = 3\n1 / i = -i\n\nComplex numbers are numbers that have both a real part and an imaginary part, and are usually\nnoted:\n\na + bi\n\nwhere \"a\" is the *real* part and \"b\" is the *imaginary* part. The arithmetic with complex\nnumbers is straightforward. You have to keep track of the real and the imaginary parts, but\notherwise the rules used for real numbers just apply:\n\n(4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i\n(2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i\n\nA graphical representation of complex numbers is possible in a plane (also called the *complex\nplane*, but it's really a 2D plane). The number\n\nz = a + bi\n\nis the point whose coordinates are (a, b). Actually, it would be the vector originating from (0,\n0) to (a, b). It follows that the addition of two complex numbers is a vectorial addition.\n\nSince there is a bijection between a point in the 2D plane and a complex number (i.e. the\nmapping is unique and reciprocal), a complex number can also be uniquely identified with polar\ncoordinates:\n\n[rho, theta]\n\nwhere \"rho\" is the distance to the origin, and \"theta\" the angle between the vector and the *x*\naxis. There is a notation for this using the exponential form, which is:\n\nrho * exp(i * theta)\n\nwhere *i* is the famous imaginary number introduced above. Conversion between this form and the\ncartesian form \"a + bi\" is immediate:\n\na = rho * cos(theta)\nb = rho * sin(theta)\n\nwhich is also expressed by this formula:\n\nz = rho * exp(i * theta) = rho * (cos theta + i * sin theta)\n\nIn other words, it's the projection of the vector onto the *x* and *y* axes. Mathematicians call\n*rho* the *norm* or *modulus* and *theta* the *argument* of the complex number. The *norm* of\n\"z\" is marked here as abs(z).\n\nThe polar notation (also known as the trigonometric representation) is much more handy for\nperforming multiplications and divisions of complex numbers, whilst the cartesian notation is\nbetter suited for additions and subtractions. Real numbers are on the *x* axis, and therefore\n*y* or *theta* is zero or *pi*.\n\nAll the common operations that can be performed on a real number have been defined to work on\ncomplex numbers as well, and are merely *extensions* of the operations defined on real numbers.\nThis means they keep their natural meaning when there is no imaginary part, provided the number\nis within their definition set.\n\nFor instance, the \"sqrt\" routine which computes the square root of its argument is only defined\nfor non-negative real numbers and yields a non-negative real number (it is an application from\nR+ to R+). If we allow it to return a complex number, then it can be extended to negative real\nnumbers to become an application from R to C (the set of complex numbers):\n\nsqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i\n\nIt can also be extended to be an application from C to C, whilst its restriction to R behaves as\ndefined above by using the following definition:\n\nsqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)\n\nIndeed, a negative real number can be noted \"[x,pi]\" (the modulus *x* is always non-negative, so\n\"[x,pi]\" is really \"-x\", a negative number) and the above definition states that\n\nsqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i\n\nwhich is exactly what we had defined for negative real numbers above. The \"sqrt\" returns only\none of the solutions: if you want the both, use the \"root\" function.\n\nAll the common mathematical functions defined on real numbers that are extended to complex\nnumbers share that same property of working *as usual* when the imaginary part is zero\n(otherwise, it would not be called an extension, would it?).\n\nA *new* operation possible on a complex number that is the identity for real numbers is called\nthe *conjugate*, and is noted with a horizontal bar above the number, or \"~z\" here.\n\nz = a + bi\n~z = a - bi\n\nSimple... Now look:\n\nz * ~z = (a + bi) * (a - bi) = a*a + b*b\n\nWe saw that the norm of \"z\" was noted abs(z) and was defined as the distance to the origin, also\nknown as:\n\nrho = abs(z) = sqrt(a*a + b*b)\n\nso\n\nz * ~z = abs(z)  2\n\nIf z is a pure real number (i.e. \"b == 0\"), then the above yields:\n\na * a = abs(a)  2\n\nwhich is true (\"abs\" has the regular meaning for real number, i.e. stands for the absolute\nvalue). This example explains why the norm of \"z\" is noted abs(z): it extends the \"abs\" function\nto complex numbers, yet is the regular \"abs\" we know when the complex number actually has no\nimaginary part... This justifies *a posteriori* our use of the \"abs\" notation for the norm.\n",
            "subsections": []
        },
        "OPERATIONS": {
            "content": "Given the following notations:\n\nz1 = a + bi = r1 * exp(i * t1)\nz2 = c + di = r2 * exp(i * t2)\nz = <any complex or real number>\n\nthe following (overloaded) operations are supported on complex numbers:\n\nz1 + z2 = (a + c) + i(b + d)\nz1 - z2 = (a - c) + i(b - d)\nz1 * z2 = (r1 * r2) * exp(i * (t1 + t2))\nz1 / z2 = (r1 / r2) * exp(i * (t1 - t2))\nz1  z2 = exp(z2 * log z1)\n~z = a - bi\nabs(z) = r1 = sqrt(a*a + b*b)\nsqrt(z) = sqrt(r1) * exp(i * t/2)\nexp(z) = exp(a) * exp(i * b)\nlog(z) = log(r1) + i*t\nsin(z) = 1/2i (exp(i * z1) - exp(-i * z))\ncos(z) = 1/2 (exp(i * z1) + exp(-i * z))\natan2(y, x) = atan(y / x) # Minding the right quadrant, note the order.\n\nThe definition used for complex arguments of atan2() is\n\n-i log((x + iy)/sqrt(x*x+y*y))\n\nNote that atan2(0, 0) is not well-defined.\n\nThe following extra operations are supported on both real and complex numbers:\n\nRe(z) = a\nIm(z) = b\narg(z) = t\nabs(z) = r\n\ncbrt(z) = z  (1/3)\nlog10(z) = log(z) / log(10)\nlogn(z, n) = log(z) / log(n)\n\ntan(z) = sin(z) / cos(z)\n\ncsc(z) = 1 / sin(z)\nsec(z) = 1 / cos(z)\ncot(z) = 1 / tan(z)\n\nasin(z) = -i * log(i*z + sqrt(1-z*z))\nacos(z) = -i * log(z + i*sqrt(1-z*z))\natan(z) = i/2 * log((i+z) / (i-z))\n\nacsc(z) = asin(1 / z)\nasec(z) = acos(1 / z)\nacot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))\n\nsinh(z) = 1/2 (exp(z) - exp(-z))\ncosh(z) = 1/2 (exp(z) + exp(-z))\ntanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))\n\ncsch(z) = 1 / sinh(z)\nsech(z) = 1 / cosh(z)\ncoth(z) = 1 / tanh(z)\n\nasinh(z) = log(z + sqrt(z*z+1))\nacosh(z) = log(z + sqrt(z*z-1))\natanh(z) = 1/2 * log((1+z) / (1-z))\n\nacsch(z) = asinh(1 / z)\nasech(z) = acosh(1 / z)\nacoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))\n\n*arg*, *abs*, *log*, *csc*, *cot*, *acsc*, *acot*, *csch*, *coth*, *acosech*, *acotanh*, have\naliases *rho*, *theta*, *ln*, *cosec*, *cotan*, *acosec*, *acotan*, *cosech*, *cotanh*,\n*acosech*, *acotanh*, respectively. \"Re\", \"Im\", \"arg\", \"abs\", \"rho\", and \"theta\" can be used\nalso as mutators. The \"cbrt\" returns only one of the solutions: if you want all three, use the\n\"root\" function.\n\nThe *root* function is available to compute all the *n* roots of some complex, where *n* is a\nstrictly positive integer. There are exactly *n* such roots, returned as a list. Getting the\nnumber mathematicians call \"j\" such that:\n\n1 + j + j*j = 0;\n\nis a simple matter of writing:\n\n$j = ((root(1, 3))[1];\n\nThe *k*th root for \"z = [r,t]\" is given by:\n\n(root(z, n))[k] = r(1/n) * exp(i * (t + 2*k*pi)/n)\n\nYou can return the *k*th root directly by \"root(z, n, k)\", indexing starting from *zero* and\nending at *n - 1*.\n\nThe *spaceship* numeric comparison operator, <=>, is also defined. In order to ensure its\nrestriction to real numbers is conform to what you would expect, the comparison is run on the\nreal part of the complex number first, and imaginary parts are compared only when the real parts\nmatch.\n",
            "subsections": []
        },
        "CREATION": {
            "content": "To create a complex number, use either:\n\n$z = Math::Complex->make(3, 4);\n$z = cplx(3, 4);\n\nif you know the cartesian form of the number, or\n\n$z = 3 + 4*i;\n\nif you like. To create a number using the polar form, use either:\n\n$z = Math::Complex->emake(5, pi/3);\n$x = cplxe(5, pi/3);\n\ninstead. The first argument is the modulus, the second is the angle (in radians, the full circle\nis 2*pi). (Mnemonic: \"e\" is used as a notation for complex numbers in the polar form).\n\nIt is possible to write:\n\n$x = cplxe(-3, pi/4);\n\nbut that will be silently converted into \"[3,-3pi/4]\", since the modulus must be non-negative\n(it represents the distance to the origin in the complex plane).\n\nIt is also possible to have a complex number as either argument of the \"make\", \"emake\", \"cplx\",\nand \"cplxe\": the appropriate component of the argument will be used.\n\n$z1 = cplx(-2,  1);\n$z2 = cplx($z1, 4);\n\nThe \"new\", \"make\", \"emake\", \"cplx\", and \"cplxe\" will also understand a single (string) argument\nof the forms\n\n2-3i\n-3i\n[2,3]\n[2,-3pi/4]\n[2]\n\nin which case the appropriate cartesian and exponential components will be parsed from the\nstring and used to create new complex numbers. The imaginary component and the theta,\nrespectively, will default to zero.\n\nThe \"new\", \"make\", \"emake\", \"cplx\", and \"cplxe\" will also understand the case of no arguments:\nthis means plain zero or (0, 0).\n",
            "subsections": []
        },
        "DISPLAYING": {
            "content": "When printed, a complex number is usually shown under its cartesian style *a+bi*, but there are\nlegitimate cases where the polar style *[r,t]* is more appropriate. The process of converting\nthe complex number into a string that can be displayed is known as *stringification*.\n\nBy calling the class method \"Math::Complex::displayformat\" and supplying either \"polar\" or\n\"cartesian\" as an argument, you override the default display style, which is \"cartesian\". Not\nsupplying any argument returns the current settings.\n\nThis default can be overridden on a per-number basis by calling the \"displayformat\" method\ninstead. As before, not supplying any argument returns the current display style for this\nnumber. Otherwise whatever you specify will be the new display style for *this* particular\nnumber.\n\nFor instance:\n\nuse Math::Complex;\n\nMath::Complex::displayformat('polar');\n$j = (root(1, 3))[1];\nprint \"j = $j\\n\";               # Prints \"j = [1,2pi/3]\"\n$j->displayformat('cartesian');\nprint \"j = $j\\n\";               # Prints \"j = -0.5+0.866025403784439i\"\n\nThe polar style attempts to emphasize arguments like *k*pi/n* (where *n* is a positive integer\nand *k* an integer within [-9, +9]), this is called *polar pretty-printing*.\n\nFor the reverse of stringifying, see the \"make\" and \"emake\".\n\nCHANGED IN PERL 5.6\nThe \"displayformat\" class method and the corresponding \"displayformat\" object method can now\nbe called using a parameter hash instead of just a one parameter.\n\nThe old display format style, which can have values \"cartesian\" or \"polar\", can be changed using\nthe \"style\" parameter.\n\n$j->displayformat(style => \"polar\");\n\nThe one parameter calling convention also still works.\n\n$j->displayformat(\"polar\");\n\nThere are two new display parameters.\n\nThe first one is \"format\", which is a sprintf()-style format string to be used for both numeric\nparts of the complex number(s). The is somewhat system-dependent but most often it corresponds\nto \"%.15g\". You can revert to the default by setting the \"format\" to \"undef\".\n\n# the $j from the above example\n\n$j->displayformat('format' => '%.5f');\nprint \"j = $j\\n\";               # Prints \"j = -0.50000+0.86603i\"\n$j->displayformat('format' => undef);\nprint \"j = $j\\n\";               # Prints \"j = -0.5+0.86603i\"\n\nNotice that this affects also the return values of the \"displayformat\" methods: in list context\nthe whole parameter hash will be returned, as opposed to only the style parameter value. This is\na potential incompatibility with earlier versions if you have been calling the \"displayformat\"\nmethod in list context.\n\nThe second new display parameter is \"polarprettyprint\", which can be set to true or false, the\ndefault being true. See the previous section for what this means.\n",
            "subsections": []
        },
        "USAGE": {
            "content": "Thanks to overloading, the handling of arithmetics with complex numbers is simple and almost\ntransparent.\n\nHere are some examples:\n\nuse Math::Complex;\n\n$j = cplxe(1, 2*pi/3);  # $j  3 == 1\nprint \"j = $j, j3 = \", $j  3, \"\\n\";\nprint \"1 + j + j2 = \", 1 + $j + $j2, \"\\n\";\n\n$z = -16 + 0*i;                 # Force it to be a complex\nprint \"sqrt($z) = \", sqrt($z), \"\\n\";\n\n$k = exp(i * 2*pi/3);\nprint \"$j - $k = \", $j - $k, \"\\n\";\n\n$z->Re(3);                      # Re, Im, arg, abs,\n$j->arg(2);                     # (the last two aka rho, theta)\n# can be used also as mutators.\n",
            "subsections": []
        },
        "CONSTANTS": {
            "content": "PI\nThe constant \"pi\" and some handy multiples of it (pi2, pi4, and pip2 (pi/2) and pip4 (pi/4)) are\nalso available if separately exported:\n\nuse Math::Complex ':pi';\n$thirdofcircle = pi2 / 3;\n",
            "subsections": [
                {
                    "name": "Inf",
                    "content": "The floating point infinity can be exported as a subroutine Inf():\n\nuse Math::Complex qw(Inf sinh);\nmy $AlsoInf = Inf() + 42;\nmy $AnotherInf = sinh(1e42);\nprint \"$AlsoInf is $AnotherInf\\n\" if $AlsoInf == $AnotherInf;\n\nNote that the stringified form of infinity varies between platforms: it can be for example any\nof\n\ninf\ninfinity\nINF\n1.#INF\n\nor it can be something else.\n\nAlso note that in some platforms trying to use the infinity in arithmetic operations may result\nin Perl crashing because using an infinity causes SIGFPE or its moral equivalent to be sent. The\nway to ignore this is\n\nlocal $SIG{FPE} = sub { };\n"
                }
            ]
        },
        "ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO": {
            "content": "The division (/) and the following functions\n\nlog     ln      log10   logn\ntan     sec     csc     cot\natan    asec    acsc    acot\ntanh    sech    csch    coth\natanh   asech   acsch   acoth\n\ncannot be computed for all arguments because that would mean dividing by zero or taking\nlogarithm of zero. These situations cause fatal runtime errors looking like this\n\ncot(0): Division by zero.\n(Because in the definition of cot(0), the divisor sin(0) is 0)\nDied at ...\n\nor\n\natanh(-1): Logarithm of zero.\nDied at...\n\nFor the \"csc\", \"cot\", \"asec\", \"acsc\", \"acot\", \"csch\", \"coth\", \"asech\", \"acsch\", the argument\ncannot be 0 (zero). For the logarithmic functions and the \"atanh\", \"acoth\", the argument cannot\nbe 1 (one). For the \"atanh\", \"acoth\", the argument cannot be -1 (minus one). For the \"atan\",\n\"acot\", the argument cannot be \"i\" (the imaginary unit). For the \"atan\", \"acoth\", the argument\ncannot be \"-i\" (the negative imaginary unit). For the \"tan\", \"sec\", \"tanh\", the argument cannot\nbe *pi/2 + k * pi*, where *k* is any integer. atan2(0, 0) is undefined, and if the complex\narguments are used for atan2(), a division by zero will happen if z12+z22 == 0.\n\nNote that because we are operating on approximations of real numbers, these errors can happen\nwhen merely `too close' to the singularities listed above.\n",
            "subsections": []
        },
        "ERRORS DUE TO INDIGESTIBLE ARGUMENTS": {
            "content": "The \"make\" and \"emake\" accept both real and complex arguments. When they cannot recognize the\narguments they will die with error messages like the following\n\nMath::Complex::make: Cannot take real part of ...\nMath::Complex::make: Cannot take real part of ...\nMath::Complex::emake: Cannot take rho of ...\nMath::Complex::emake: Cannot take theta of ...\n",
            "subsections": []
        },
        "BUGS": {
            "content": "Saying \"use Math::Complex;\" exports many mathematical routines in the caller environment and\neven overrides some (\"sqrt\", \"log\", \"atan2\"). This is construed as a feature by the Authors,\nactually... ;-)\n\nAll routines expect to be given real or complex numbers. Don't attempt to use BigFloat, since\nPerl has currently no rule to disambiguate a '+' operation (for instance) between two overloaded\nentities.\n\nIn Cray UNICOS there is some strange numerical instability that results in root(), cos(), sin(),",
            "subsections": [
                {
                    "name": "cosh",
                    "content": "compiler, in Math::Complex. Whatever it is, it does not manifest itself anywhere else where Perl\nruns.\n"
                }
            ]
        },
        "SEE ALSO": {
            "content": "Math::Trig\n",
            "subsections": []
        },
        "AUTHORS": {
            "content": "Daniel S. Lewart <lewart!at!uiuc.edu>, Jarkko Hietaniemi <jhi!at!iki.fi>, Raphael Manfredi\n<RaphaelManfredi!at!pobox.com>, Zefram <zefram@fysh.org>\n",
            "subsections": []
        },
        "LICENSE": {
            "content": "This library is free software; you can redistribute it and/or modify it under the same terms as\nPerl itself.\n",
            "subsections": []
        }
    },
    "summary": "Math::Complex - complex numbers and associated mathematical functions",
    "flags": [],
    "examples": [],
    "see_also": []
}