IRIX Base Documentation 2001 May/SGI IRIX Base Documentation 2001 May.iso/usr/share/catman/p_man/cat3/perl5/Math::Complex.z/Math::Complex

Text File | 1998-10-30 | 17.5 KB | 529 lines

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     Math::Complex - complex numbers and associated mathematical functions

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             use Math::Complex;

             $z = Math::Complex->make(5, 6);
             $t = 4 - 3*i + $z;
             $j = cplxe(1, 2*pi/3);


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     This package lets you create and manipulate complex numbers. By default,
     _P_e_r_l 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:

             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 _i_m_a_g_i_n_a_r_y number.

     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 the _r_e_a_l part and b is the _i_m_a_g_i_n_a_r_y 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:

             (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




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     A graphical representation of complex numbers is possible in a plane
     (also called the _c_o_m_p_l_e_x _p_l_a_n_e, 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 the _x axis. There is a notation for this using the exponential
     form, which is:

             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:

             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 the _x and _y axes.
     Mathematicians call _r_h_o the _n_o_r_m or _m_o_d_u_l_u_s and _t_h_e_t_a the _a_r_g_u_m_e_n_t of the
     complex number. The _n_o_r_m of 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 the _x axis, and therefore _t_h_e_t_a is zero
     or _p_i.

     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
     _e_x_t_e_n_s_i_o_n_s 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 RRRR++++ to RRRR++++).  If we allow
     it to return a complex number, then it can be extended to negative real
     numbers to become an application from RRRR to CCCC (the set of complex



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     numbers):

             sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i

     It can also be extended to be an application from CCCC to CCCC, whilst its
     restriction to RRRR behaves 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 modulus _x is
     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.

     All the common mathematical functions defined on real numbers that are
     extended to complex numbers share that same property of working _a_s _u_s_u_a_l
     when the imaginary part is zero (otherwise, it would not be called an
     extension, would it?).

     A _n_e_w operation possible on a complex number that is the identity for
     real numbers is called the _c_o_n_j_u_g_a_t_e, 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 justifies _a _p_o_s_t_e_r_i_o_r_i our use of the abs notation for the



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     norm.

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     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)
             ~z1 = a - bi
             abs(z1) = r1 = sqrt(a*a + b*b)
             sqrt(z1) = sqrt(r1) * exp(i * t1/2)
             exp(z1) = exp(a) * exp(i * b)
             log(z1) = log(r1) + i*t1
             sin(z1) = 1/2i (exp(i * z1) - exp(-i * z1))
             cos(z1) = 1/2 (exp(i * z1) + exp(-i * z1))
             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

             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))






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             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))

     _l_o_g, _c_s_c, _c_o_t, _a_c_s_c, _a_c_o_t, _c_s_c_h, _c_o_t_h, _a_c_o_s_e_c_h, _a_c_o_t_a_n_h, have aliases _l_n,
     _c_o_s_e_c, _c_o_t_a_n, _a_c_o_s_e_c, _a_c_o_t_a_n, _c_o_s_e_c_h, _c_o_t_a_n_h, _a_c_o_s_e_c_h, _a_c_o_t_a_n_h,
     respectively.

     The _r_o_o_t 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:

             1 + j + j*j = 0;

     is a simple matter of writing:

             $j = ((root(1, 3))[1];

     The _kth root for z = [r,t] is given by:

             (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)

     The _s_p_a_c_e_s_h_i_p 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.

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     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:




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             $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).

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     When printed, a complex number is usually shown under its cartesian form
     _a+_b_i, 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 for _t_h_i_s particular 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 like _k*_p_i/_n (where _n is
     a positive integer and _k an integer within [-9,+9]).

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     Thanks to overloading, the handling of arithmetics with complex numbers
     is simple and almost transparent.

     Here are some examples:

             use Math::Complex;






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             $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";


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     The division (/) and the following functions

             tan
             sec
             csc
             cot
             asec
             acsc
             atan
             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 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, sech, the argument cannot be _p_i/_2 + _k * _p_i, where
     _k is any integer.




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BBBBUUUUGGGGSSSS
     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.

AAAAUUUUTTTTHHHHOOOORRRRSSSS
     Raphael Manfredi <_R_a_p_h_a_e_l__M_a_n_f_r_e_d_i@_g_r_e_n_o_b_l_e._h_p._c_o_m> and Jarkko Hietaniemi
     <_j_h_i@_i_k_i._f_i>.

     Extensive patches by Daniel S. Lewart <_d-_l_e_w_a_r_t@_u_i_u_c._e_d_u>.









































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