Exp
Exp[z]
gives the exponential of z.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For certain special arguments, Exp automatically evaluates to exact values.
- Exp can be evaluated to arbitrary numerical precision.
- Exp automatically threads over lists.
- Exp[z] is converted to E^z.
- Exp can be used with Interval and CenteredInterval objects. »
Examples
open allclose allBasic Examples (6)Summary of the most common use cases
Scope (55)Survey of the scope of standard use cases
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Exp can take complex number inputs:
Evaluate Exp efficiently at high precision:
Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:
Or compute average-case statistical intervals using Around:
Compute the elementwise values of an array:
Or compute the matrix Exp function using MatrixFunction:
Specific Values (6)
Visualization (4)
Function Properties (12)
Exp is defined for all real and complex values:
Exp achieves all positive values on the reals:
The range for complex values is the entire plane except for 0:
Exp is a periodic function with period :
Exp has the mirror property :
Exp is an analytic function of x:
Exp is non-decreasing:
Exp is injective:
Exp is not surjective:
Exp is non-negative:
Has no singularities or discontinuities:
Exp is convex:
TraditionalForm formatting:
Differentiation (3)
Integration (5)
Series Expansions (5)
Integral Transforms (3)
Function Identities and Simplifications (6)
Function Representations (5)
Exp arises from the power function in a limit:
Representation in terms of Bessel functions:
Exp can be represented in terms of MeijerG:
Exp can be represented as a DifferentialRoot:
Applications (15)Sample problems that can be solved with this function
Differential Equations (7)
Solution of a boundary‐layer problem using Exp:
Calculate the dispersion relation for the telegrapher's equation using a plane wave ansatz:
Solve the Schrödinger equation for the exponential Liouville potential:
Transmission and reflection coefficient of the Schrödinger equation for a step potential:
Propagator for the free‐particle Schrödinger equation:
Probability, Statistics and Statistical Mechanics (4)
Define the CDF of the Gumbel distribution through nested exponential functions:
Calculate the first moment symbolically:
Define a Fermi–Dirac, a Bose–Einstein and a Maxwell–Boltzmann distribution function:
Calculate the moments of the binomial distribution from the exponential generating function:
Limits and Expansions (2)
Take this multivariate function:
Find series solution up to order three for the following system of equations:
The result satisfies the equations:
Construct a fast growing function using Exp and compute its limit:
Properties & Relations (19)Properties of the function, and connections to other functions
Convert from exponential to trigonometric and hyperbolic functions:
Convert trigonometric and hyperbolic functions into exponentials:
Calculate special values as radicals:
Extract numerators and denominators:
Reciprocals of the exponential function evaluate to exponential functions:
Exp arises from the power function in a limit:
Compose with inverse functions:
PowerExpand disregards multivaluedness of Log:
Obtain a form correct for all complex ‐values:
Compose with inverse trigonometric and hyperbolic functions:
Solve transcendental equations involving Exp:
Reduce an exponential equation:
The coefficients of the series of nested exponential functions are multiples of Bell numbers:
Exp is a numeric function:
The generating function for Exp:
FindSequenceFunction can recognize the Exp sequence:
The exponential generating function for Exp:
Possible Issues (7)Common pitfalls and unexpected behavior
Exponentials can be very large:
And can become too large for computer representation of a number:
Literal matchings may fail because exponential functions evaluate to powers with base E:
Use Unevaluated or Hold to avoid evaluation:
Logarithms in exponents are not always automatically resolved:
Use Together to remove logarithms in exponents:
Machine-precision input is insufficient to give a correct answer:
With exact input, the answer is correct:
No power series exists at infinity, where Exp has an essential singularity:
Exp is applied elementwise to matrices; MatrixExp finds matrix exponentials:
In TraditionalForm input, parentheses are needed around the argument:
Neat Examples (5)Surprising or curious use cases
Find correction terms to a classic limit:
Closed-form expression for the partial sum of the power series of Exp:
Leading correction for the difference to Exp[z] for large :
Nested exponential functions over the complex plane:
Fractal from iterating Exp:
The almost nowhere differentiable Riemann–Weierstrass function:
Text
Wolfram Research (1988), Exp, Wolfram Language function, https://reference.wolfram.com/language/ref/Exp.html (updated 2021).
CMS
Wolfram Language. 1988. "Exp." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Exp.html.
APA
Wolfram Language. (1988). Exp. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Exp.html