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Exp

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Exp[z]
    gives the exponential E^z.
Exp is Listable. Exp[0] = 1, Exp[Log[x]] -> x, Exp[I Pi] = -1.
Numeric inputs route to libm / MPFR.

Examples

No verified examples yet for this function.

Implementation notes

Algorithm. builtin_exp (1-arg). Exact special values: Exp[0] = 1, Exp[-Infinity] = 0, Exp[Infinity] = Infinity. The notable closed-form path is Euler's formula for Exp[I q Pi]: when the argument is Times[Complex[0,q], Pi] with q an integer or rational, it rewrites to Cos[q Pi] + I Sin[q Pi] (which the trig kernels then evaluate to exact algebraic values). Numeric arguments go through MPFR when USE_MPFR — mpfr_exp for pure reals, the exp(a)(cos b + i sin b) complex helper otherwise — or cexp on a double complex for machine inputs, collapsing to a real result when the imaginary part is zero. Anything that matches none of these is returned as Power[E, z] (the canonical internal form for an unevaluated exponential), so Exp is effectively a thin front-end that produces E^z and lets the Power machinery carry the symbolic case.

Data structures. Expr* trees throughout; numeric evaluation uses double complex (machine) or mpfr_t/complex-MPFR helpers (arbitrary precision). The Euler path scans the Times argument list for a Pi factor and a single pure-imaginary Complex[0, q] coefficient.

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Exp[0]
Out[1]= 1

In[2]:= Exp[I Pi]
Out[2]= -1

In[3]:= Exp[Log[x]]
Out[3]= x

In[1]:= Exp[2 Log[x]]
Out[1]= x^2

In[1]:= D[Exp[Sin[x]], x]
Out[1]= Cos[x] E^Sin[x]

In[1]:= Series[Exp[x], {x, 0, 6}]
Out[1]= 1 + x + 1/2 x^2 + 1/6 x^3 + 1/24 x^4 + 1/120 x^5 + 1/720 x^6 + O[x]^7

In[1]:= N[Exp[1], 50]
Out[1]= 2.71828182845904523536028747135266249775724709369996

Notes

Exp[z] is E^z; it inverts Log and evaluates Euler's identity exactly. Logarithmic arguments collapse (Exp[2 Log[x]] -> x^2), it differentiates and series-expands symbolically, and numeric inputs route to libm / MPFR for arbitrary precision. Exp is Listable.