Reference Guide

3-20 Full Command and Function Reference
Input: Level 1/Item 1: An expression giving the name of the global variable to be added to the
REALASSUME list, and the assumption to be placed on it, or a list of such assumptions.
Output: Level 1/Item 1: The input expression or list of expressions.
Example: Add the CAS assumption that the global variable Z is real and positive. Note that ASSUME will
replace Z>0 with Z≥0, which does not guarantee that Z is positive, so Z≥MINR is used instead,
which guarantees that Z is greater than or equal to the smallest positive number the calculator
recognizes.
Command: ASSUME(Z≥MINR)
Result: Z≥MINR
See also: ADDTOREAL, UNASSUME
ATAN
Type: Analytic Function
Description: Arc Tangent Analytic Function: Returns the value of the angle having the given tangent.
For a real argument, the result ranges from –90 to +90 degrees (–π/2 to +π/2 radians; –100 to
+100 grads).
The inverse of TAN is a relation, not a function, since TAN sends more than one argument to the
same result. The inverse relation for TAN is expressed by ISOL as the general solution:
ATAN(Z)+π*n1
The function ATAN is the inverse of a part of TAN, a part defined by restricting the domain of
TAN such that:
• each argument is sent to a distinct result, and
• each possible result is achieved.
The points in this restricted domain of TAN are called the principal values of the inverse relation.
ATAN in its entirety is called the principal branch of the inverse relation, and the points sent by
ATAN to the boundary of the restricted domain of TAN form the branch cuts of ATAN.
The principal branch used by the calculator for ATAN was chosen because it is analytic in the
regions where the arguments of the real-valued inverse function are defined. The branch cuts for
the complex-valued arc tangent function occur where the corresponding real-valued function is
undefined. The principal branch also preserves most of the important symmetries.
The graphs below show the domain and range of ATAN. The graph of the domain shows where
the branch cuts occur: the heavy solid line marks one side of a cut, while the feathered lines mark
the other side of a cut. The graph of the range shows where each side of each cut is mapped
under the function.
These graphs show the inverse relation ATAN(Z)+π*n1 for the case n1 = 0. For other values of
n1, the vertical band in the lower graph is translated to the right (for n1 positive) or to the left (for
n1 negative). Together, the bands cover the whole complex plane, the domain of TAN.
View these graphs with domain and range reversed to see how the domain of TAN is restricted to
make an inverse function possible. Consider the vertical band in the lower graph as the restricted
domain Z = (x, y). TAN sends this domain onto the whole complex plane in the range W = (u,
v) = TAN(x, y) in the upper graph.