Sollya User's Manual

Name:

^ power function

Usage:

function1 ^ function2 : (function, function) -> function interval1 ^ interval2 : (range, range) -> range interval1 ^ constant : (range, constant) -> range interval1 ^ constant : (constant, range) -> range

Parameters:

function1 and function2 represent functions
interval1 and interval2 represent intervals (ranges)
constant represents a constant or constant expression

Description:

^ represents the power (function) on reals. The expression function1 ^ function2 stands for the function composed of the power function and the two functions function1 and function2, where function1 is the base and function2 the exponent. If function2 is a constant integer, ^ is defined on negative values of function1. Otherwise ^ is defined as exp(y * log(x)).
Note that whenever several ^ are composed, the priority goes to the last ^. This corresponds to the natural way of thinking when a tower of powers is written on a paper. Thus, 2^3^5 is interpreted as 2^(3^5).
^ can be used for interval arithmetic on intervals (ranges). ^ will evaluate to an interval that safely encompasses all images of the power function with arguments varying in the given intervals. If the intervals given contain points where the power function is not defined, infinities and NaNs will be produced in the output interval. Any combination of intervals with intervals or constants (resp. constant expressions) is supported. However, it is not possible to represent families of functions using an interval as one argument and a function (varying in the free variable) as the other one.

Example 1:

> 5 ^ 2;
25

Example 2:

> x ^ 2;
x^2

Example 3:

> 3 ^ (-5);
4.1152263374485596707818930041152263374485596707818e-3

Example 4:

> (-3) ^ (-2.5);
@NaN@

Example 5:

> diff(sin(x) ^ exp(x));
sin(x)^exp(x) * ((cos(x) * exp(x)) / sin(x) + exp(x) * log(sin(x)))

Example 6:

   > 2^3^5;
   1.4134776518227074636666380005943348126619871175005e73
   > (2^3)^5;
   32768
   > 2^(3^5);
   1.4134776518227074636666380005943348126619871175005e73

Example 7:

   > [1;2] ^ [3;4];
   [1;1.60000000000000000000000000000000000000000000000007e1]
   > [1;2] ^ 17;
   [1;131072]
   > 13 ^ [-4;17];
   [3.501277966457757081334687160813696999404782745702e-5;8650415919381337933]

WARNING