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.

Examples

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Basic Examples (6)

Evaluate numerically:

Evaluate numerically to any precision:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Exponential functions can be entered as ee x:

Scope (48)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Exp can take complex number inputs:

Evaluate Exp efficiently at high precision:

Exp can deal with real‐valued intervals:

Exp threads elementwise over lists and matrices:

Specific Values (6)

The value at zero:

Values of Exp at fixed points:

Values at infinity:

Simple exact values are generated automatically:

Some more complicated values can be expanded using ExpToTrig:

Local extrema of Exp along the imaginary axis:

Find a value of for which the using Solve:

Substitute in the result:

Visualize the result:

Visualization (4)

Plot the Exp function:

Plot the real and imaginary parts of Exp[I x]:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

Function Properties (5)

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 :

TraditionalForm formatting:

Differentiation (3)

First derivative:

Formula for the derivative:

Derivative of a nested exponential function:

Integration (5)

Indefinite integral of Exp:

Definite integral of Exp:

Gaussian integral:

Gamma function definition:

More integrals:

Series Expansions (5)

Taylor expansion for Exp:

Plot the first three approximations for Exp around :

General term in the series expansion of Exp:

Series expansion of the exponential function at infinity:

The first-order Fourier series:

Exp can be applied to power series:

Integral Transforms (3)

Compute the Fourier transforms using FourierTransform:

LaplaceTransform:

MellinTransform:

Function Identities and Simplifications (6)

Primary definition:

Euler's formula:

Convert from exponential to hyperbolic functions:

Convert trigonometric and hyperbolic functions into exponentials:

Products are automatically combined:

Expand assuming real variables x and y:

Function Representations (5)

Exp arises from the power function in a limit:

Series representation:

Representation in terms of Bessel functions:

Exp can be represented in terms of MeijerG:

Exp can be represented as a DifferentialRoot:

Applications (12)

Exponential decay:

Damped harmonic oscillator:

Normal distribution:

Calculate moments:

Define the CDF of the Gumbel distribution through nested exponential functions:

Plot the PDF:

Calculate the first moment symbolically:

Solution of a boundary‐layer problem using Exp:

Plot various solutions:

Multivariate Gaussian integrals:

Calculate the dispersion relation for the telegrapher's equation using a plane wave ansatz:

Define a Fermi–Dirac, a Bose–Einstein, and a Maxwell–Boltzmann distribution function:

Plot the distributions:

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:

Calculate spreading of a Gaussian wave packet:

Visualize the spreading:

Calculate the moments of the binomial distribution from the exponential generating function:

Properties & Relations (19)

Convert from Exp to Power:

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:

Integrals:

Integral transform:

Sums:

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)

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 element‐wise to matrices; MatrixExp finds matrix exponentials:

In traditional form, parentheses are needed around the argument:

Neat Examples (5)

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:

Introduced in 1988

(1.0)

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Exp