The rules used to simplify expressions involving the complex variable functions take into account the current branch selection (see the Branch fieldBranch_field of the Options > Mode Settings > Simplification command19_L5FP) and the user-declared domain of variables (see the Author > Variable Domain command8MNT4Y).


#i is the imaginary unit, SQRT(-1).  #i displays as î, and can be entered by clicking on the î on the math symbol toolbar.  You are free to use the unembellished variable name i to represent electrical current, interest, etc.


unit_circle represents some arbitrary point on the unit circle in the complex plane.  For instance, 1, -1, î, and -î are points on the unit circle.  unit_circle may arise in limit problems or when solving equations.  For example, if z is declared complex (see the Author > Variable Domain command8MNT4Y), solving the equation |z| = 2 for z gives z = 2·unit_circle.


ABS(z) simplifies to the absolute value (also called magnitude or modulus) of z.  The absolute value of z is the distance between z and the origin of the complex plane.  An absolute value can be entered and is displayed using vertical bars to delimit the argument.  Hence if x and y are real,

|x + î·y|

simplifies to

   2    2 
√(x  + y )


SIGN(z) simplifies to the point on the unit circle in the complex plane that has the same phase angle as z.  For example,

SIGN(3 + 4·î)

simplifies to

 3     4·î 
——— + —————
 5      5  


RE(z) simplifies to the real part of z.  Hence if x and y are real, RE(x + îy) simplifies to x.


IM(z) simplifies to the imaginary part of z.  Hence if x and y are real, IM(x + îy) simplifies to y.


CONJ(z) simplifies to the complex conjugate of z.  Hence if x and y are real, CONJ(x+îy) simplifies to x - îy.


PHASE(z) simplifies to the principal phase angle of z, measured counterclockwise from the positive x-axis.  The angle is displayed in the angular unit (degrees or radians) specified by the Angular Unit fieldAngular_Unit_field of the Options > Mode Settings > Simplification command19_L5FP.  Thus if the angular unit is a radian, PHASE(z) lies in the interval (-π, π] for any z in the complex plane.


Other Built-in Functions and ConstantsBuilt_in_Functions_and_Constants 

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