For x = 0 or (0, 0), an Infinite Result exception occurs, or, if flag –22 is set, –MAXR is returned. The inverse of EXP is a relation, not a function, since EXP sends more than one argument to the same result. The inverse relation for EXP is the general solution:

LN(Z)+2*π*i*n1

The function LN is the inverse of a part of EXP, a part defined by restricting the domain of EXP such that: each argument is sent to a distinct result, and each possible result is achieved.

The points in this restricted domain of EXP are called the principal values of the inverse relation. LN in its entirety is called the principal branch of the inverse relation, and the points sent by LN to the boundary of the restricted domain of EXP form the branch cuts of LN.

The principal branch used by the calculator for LN was chosen because it is analytic in the regions where the arguments of the real-valuedinverse function are defined. The branch cut for the complex-valued natural log function occurs 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 LN. The graph of the domain shows where the branch cut occurs: the heavy solid line marks one side of the cut, while the feathered lines mark the other side of the cut. The graph of the range shows where each side of the cut is mapped under the function.

These graphs show the inverse relation LN(Z)+2*π*i*n1 for the case n1=0. For other values of n1, the horizontal band in the lower graph is translated up (for n1 positive) or down (for n1 negative). Taken together, the bands cover the whole complex plane, which is the domain of EXP.

You can view these graphs with domain and range reversed to see how the domain of EXP is restricted to make an inverse function possible. Consider the vertical band in the lower graph as the restricted domain Z = (x,y). EXP sends this domain onto the whole complex plane in the range W = (u,v) = EXP(x,y) in the upper graph.

Access: …¹

(¹is the right-shift of the Qkey).

Flags: Principal Solution (–1), Numerical Results (–3), Infinite Result Exception (–22)

Full Command and Function Reference 3-135