# 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(55)

### 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 realvalued 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(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:

### 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:

### 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 boundarylayer 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 FermiDirac, a BoseEinstein, and a MaxwellBoltzmann 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 freeparticle Schrödinger equation:

Calculate spreading of a Gaussian wave packet:

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:

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 elementwise 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 RiemannWeierstrass function: