Syntax:

exp number => result

expt base-number power-number => result

Arguments and Values:

number---a number.

base-number---a number.

power-number---a number.

result---a number.

Description:

exp and expt perform exponentiation.

exp returns e raised to the power number, where e is the base of the natural logarithms. exp has no branch cut.

expt returns base-number raised to the power power-number. If the base-number is a rational and power-number is an integer, the calculation is exact and the result will be of type rational; otherwise a floating-point approximation might result. For expt of a complex rational to an integer power, the calculation must be exact and the result is of type (or rational (complex rational)).

The result of expt can be a complex, even when neither argument is a complex, if base-number is negative and power-number is not an integer. The result is always the principal complex value. For example, (expt -8 1/3) is not permitted to return -2, even though -2 is one of the cube roots of -8. The principal cube root is a complex approximately equal to #C(1.0 1.73205), not -2.

expt is defined as b^x = e^x log b. This defines the principal values precisely. The range of expt is the entire complex plane. Regarded as a function of x, with b fixed, there is no branch cut. Regarded as a function of b, with x fixed, there is in general a branch cut along the negative real axis, continuous with quadrant II. The domain excludes the origin. By definition, 0^0=1. If b=0 and the real part of x is strictly positive, then b^x=0. For all other values of x, 0^x is an error.

When power-number is an integer 0, then the result is always the value one in the type of base-number, even if the base-number is zero (of any type). That is:

 (expt x 0) ==  (coerce 1 (type-of x))

Examples:

 (exp 0) =>  1.0
 (exp 1) =>  2.718282
 (exp (log 5)) =>  5.0 
 (expt 2 8) =>  256
 (expt 4 .5) =>  2.0
 (expt #c(0 1) 2) =>  -1
 (expt #c(2 2) 3) =>  #C(-16 16)
 (expt #c(2 2) 4) =>  -64 

Affected By: None.

Exceptional Situations: None.

See Also:

log, Section 12.1.3.3 (Rule of Float Substitutability)

Notes:

Implementations of expt are permitted to use different algorithms for the cases of a power-number of type rational and a power-number of type float.

Note that by the following logic, (sqrt (expt x 3)) is not equivalent to (expt x 3/2).

 (setq x (exp (/ (* 2 pi #c(0 1)) 3)))         ;exp(2.pi.i/3)
 (expt x 3) =>  1 ;except for round-off error
 (sqrt (expt x 3)) =>  1 ;except for round-off error
 (expt x 3/2) =>  -1 ;except for round-off error

• EXPT-RATIO:P.211
• COMPLEX-RATIONAL-RESULT:EXTEND