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

open allclose all

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 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  (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 exp(TemplateBox[{z}, Conjugate])=TemplateBox[{{exp, (, z, )}}, Conjugate]:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Formula for the ^(th) 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 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:

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 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:

Introduced in 1988
 (1.0)