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Calculates the absolute value of an expression representing a complex
number. Unlike the function abs, the cabs function always
decomposes its argument into a real and an imaginary part. If x and
y represent real variables or expressions, the cabs function
calculates the absolute value of x + %i*y as
(%i1) cabs (1); (%o1) 1
(%i2) cabs (1 + %i); (%o2) sqrt(2)
(%i3) cabs (exp (%i)); (%o3) 1
(%i4) cabs (exp (%pi * %i)); (%o4) 1
(%i5) cabs (exp (3/2 * %pi * %i)); (%o5) 1
(%i6) cabs (17 * exp (2 * %i)); (%o6) 17
If cabs returns a noun form this most commonly is caused by
some properties of the variables involved not being known:
(%i1) cabs (a+%i*b);
2 2
(%o1) sqrt(b + a )
(%i2) declare(a,real,b,real); (%o2) done
(%i3) cabs (a+%i*b);
2 2
(%o3) sqrt(b + a )
(%i4) assume(a>0,b>0); (%o4) [a > 0, b > 0]
(%i5) cabs (a+%i*b);
2 2
(%o5) sqrt(b + a )
The cabs function can use known properties like symmetry properties of
complex functions to help it calculate the absolute value of an expression. If
such identities exist, they can be advertised to cabs using function
properties. The symmetries that cabs understands are: mirror symmetry,
conjugate function and complex characteristic.
cabs is a verb function and is not suitable for symbolic
calculations. For such calculations (including integration,
differentiation and taking limits of expressions containing absolute
values), use abs.
The result of cabs can include the absolute value function,
abs, and the arc tangent, atan2.
When applied to a list or matrix, cabs automatically distributes over
the terms. Similarly, it distributes over both sides of an equation.
For further ways to compute with complex numbers, see the functions
rectform, realpart, imagpart,
carg, conjugate and polarform.
Examples:
(%i1) cabs(sqrt(1+%i*x));
2 1/4
(%o1) (x + 1)
(%i2) cabs(sin(x+%i*y));
2 2 2 2
(%o2) sqrt(cos (x) sinh (y) + sin (x) cosh (y))
The error function, erf, has mirror symmetry, which is used here in
the calculation of the absolute value with a complex argument:
(%i3) cabs(erf(x+%i*y));
2
(erf(%i y + x) - erf(%i y - x))
(%o3) sqrt(--------------------------------
4
2
(erf(%i y + x) + erf(%i y - x))
- --------------------------------)
4
Maxima knows complex identities for the Bessel functions, which allow
it to compute the absolute value for complex arguments. Here is an
example for bessel_j.
(%i4) cabs(bessel_j(1,%i)); (%o4) abs(bessel_j(1, %i))
Returns the complex argument of z. The complex argument is an angle
theta in (-%pi, %pi] such that r exp (theta %i) = z
where r is the magnitude of z.
carg is a computational function, not a simplifying function.
See also abs (complex magnitude), polarform,
rectform, realpart, and imagpart.
Examples:
(%i1) carg (1); (%o1) 0
(%i2) carg (1 + %i);
%pi
(%o2) ---
4
(%i3) carg (exp (%i));
sin(1)
(%o3) atan(------)
cos(1)
(%i4) carg (exp (%pi * %i)); (%o4) %pi
(%i5) carg (exp (3/2 * %pi * %i));
%pi
(%o5) - ---
2
(%i6) carg (17 * exp (2 * %i));
sin(2)
(%o6) atan(------) + %pi
cos(2)
If carg returns a noun form this most commonly is caused by
some properties of the variables involved not being known:
(%i1) carg (a+%i*b); (%o1) atan2(b, a)
(%i2) declare(a,real,b,real); (%o2) done
(%i3) carg (a+%i*b); (%o3) atan2(b, a)
(%i4) assume(a>0,b>0); (%o4) [a > 0, b > 0]
(%i5) carg (a+%i*b);
b
(%o5) atan(-)
a
Returns the complex conjugate of x.
(%i1) declare ([aa, bb], real, cc, complex, ii, imaginary); (%o1) done
(%i2) conjugate (aa + bb*%i); (%o2) aa - %i bb
(%i3) conjugate (cc); (%o3) conjugate(cc)
(%i4) conjugate (ii); (%o4) - ii
(%i5) conjugate (xx + yy); (%o5) yy + xx
Returns the imaginary part of the expression expr.
imagpart is a computational function, not a simplifying function.
See also abs, carg, polarform,
rectform, and realpart.
Example:
(%i1) imagpart (a+b*%i); (%o1) b
(%i2) imagpart (1+sqrt(2)*%i); (%o2) sqrt(2)
(%i3) imagpart (1); (%o3) 0
(%i4) imagpart (sqrt(2)*%i); (%o4) sqrt(2)
Returns an expression r %e^(%i theta) equivalent to expr,
such that r and theta are purely real.
Example:
(%i1) polarform(a+b*%i);
2 2 %i atan2(b, a)
(%o1) sqrt(b + a ) %e
(%i2) polarform(1+%i);
%i %pi
------
4
(%o2) sqrt(2) %e
(%i3) polarform(1+2*%i);
%i atan(2)
(%o3) sqrt(5) %e
Returns the real part of expr. realpart and imagpart will
work on expressions involving trigonometric and hyperbolic functions,
as well as square root, logarithm, and exponentiation.
Example:
(%i1) realpart (a+b*%i); (%o1) a
(%i2) realpart (1+sqrt(2)*%i); (%o2) 1
(%i3) realpart (sqrt(2)*%i); (%o3) 0
(%i4) realpart (1); (%o4) 1
Returns an expression a + b %i equivalent to expr,
such that a and b are purely real.
Example:
(%i1) rectform(sqrt(2)*%e^(%i*%pi/4)); (%o1) %i + 1
(%i2) rectform(sqrt(b^2+a^2)*%e^(%i*atan2(b, a))); (%o2) %i b + a
(%i3) rectform(sqrt(5)*%e^(%i*atan(2))); (%o3) 2 %i + 1
Next: Combinatorial Functions, Previous: Functions for Numbers, Up: Elementary Functions [Contents][Index]