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JacobiAmplitude
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JacobiAmplitude

✖

gives the amplitude TemplateBox[{u, m}, JacobiAmplitude] for Jacobi elliptic functions.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • JacobiAmplitude[u,m] converts from the argument u for an elliptic function to the amplitude ϕ.
  • JacobiAmplitude is the inverse of the elliptic integral of the first kind. If u=TemplateBox[{phi, m}, EllipticF], then phi=TemplateBox[{u, m}, JacobiAmplitude].
  • JacobiAmplitude[u,m] has a branch cut discontinuity in the complex u plane running from 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+1) TemplateBox[{{1, -, m}}, EllipticK] to 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+3) TemplateBox[{{1, -, m}}, EllipticK] for every pair of integers and .
  • For certain special arguments, JacobiAmplitude automatically evaluates to exact values.
  • JacobiAmplitude can be evaluated to arbitrary numerical precision.
  • JacobiAmplitude automatically threads over lists.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-kzd066
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-wm2qg
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-kiedlx
Out[1]=1

Series expansions about the origin:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-n1ltu7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-eurjnp
Out[2]=2

Scope  (26)Survey of the scope of standard use cases

Numerical Evaluation  (5)

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-o5otj
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-gwui8b
Out[2]=2

Evaluate for complex arguments:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-debjig
Out[1]=1

Evaluate JacobiAmplitude efficiently at high precision:

In[1]:=1
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https://wolfram.com/xid/0dc3l0sh425c-di5gcr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3l0sh425c-bq2c6r
Out[2]=2

Compute average-case statistical intervals using Around:

In[1]:=1
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https://wolfram.com/xid/0dc3l0sh425c-cw18bq
Out[1]=1

Compute the elementwise values of an array:

In[1]:=1
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https://wolfram.com/xid/0dc3l0sh425c-thgd2
Out[1]=1

Or compute the matrix JacobiAmplitude function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-o5jpo
Out[2]=2

Specific Values  (4)

Simple exact values are generated automatically:

In[1]:=1
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https://wolfram.com/xid/0dc3l0sh425c-dfty6e
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-kedmi
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dc3l0sh425c-crwrl3
Out[3]=3

Values at infinity for some special cases:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-cw39qs
Out[1]=1

Find a root of TemplateBox[{x, {-, 3}}, JacobiAmplitude]⩵1:

In[1]:=1
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https://wolfram.com/xid/0dc3l0sh425c-f2hrld
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3l0sh425c-ghcykx
Out[2]=2

Parity transformation is automatically applied:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-hvgb9s
Out[1]=1

Visualization  (3)

Plot the JacobiAmplitude functions for various values of parameter :

In[5]:=5
✖
https://wolfram.com/xid/0dc3l0sh425c-ecj8m7
Out[5]=5

Plot JacobiAmplitude as a function of its parameter :

In[1]:=1
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https://wolfram.com/xid/0dc3l0sh425c-du62z6
Out[1]=1

Plot the real part of TemplateBox[{z, 2}, JacobiAmplitude]:

In[1]:=1
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https://wolfram.com/xid/0dc3l0sh425c-bsucaf
Out[1]=1

Plot the imaginary part of TemplateBox[{z, 2}, JacobiAmplitude]:

In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-fpc0f
Out[2]=2

Function Properties  (5)

TemplateBox[{x, {2, /, 3}}, JacobiAmplitude] is an analytic function of x:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-h5x4l2
Out[1]=1

It has no singularities or discontinuities:

In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-mdtl3h
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dc3l0sh425c-mn5jws
Out[3]=3

TemplateBox[{x, {2, /, 3}}, JacobiAmplitude] is injective:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-poz8g
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc3l0sh425c-ctca0g
Out[2]=2

TemplateBox[{x, {2, /, 3}}, JacobiAmplitude] is surjective:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-kojhy0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-hdm869
Out[2]=2

JacobiAmplitude is neither non-negative nor non-positive:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-84dui
Out[1]=1

JacobiAmplitude is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-8kku21
Out[1]=1

Differentiation  (3)

First derivative:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-mmas49
Out[1]=1

Higher derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-nfbe0l
Out[1]=1

Plot higher derivatives for :

In[2]:=2
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https://wolfram.com/xid/0dc3l0sh425c-fxwmfc
Out[2]=2

Derivative with respect to :

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-clw10k
Out[1]=1

Series Expansions  (3)

Taylor expansion for TemplateBox[{x, {2, /, 3}}, JacobiAmplitude]:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-ewr1h8
Out[1]=1

Plot the first three approximations for TemplateBox[{x, {2, /, 3}}, JacobiAmplitude] around :

In[3]:=3
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https://wolfram.com/xid/0dc3l0sh425c-binhar
Out[4]=4

Taylor expansion for TemplateBox[{1, m}, JacobiAmplitude]:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-c7itxf
Out[1]=1

Plot the first three approximations for TemplateBox[{1, m}, JacobiAmplitude] around :

In[2]:=2
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https://wolfram.com/xid/0dc3l0sh425c-jkkunh
Out[2]=2

JacobiAmplitude can be applied to a power series:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-j9mvrp
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-bg4epz
Out[2]=2

Function Representations  (3)

JacobiAmplitude is the inverse of EllipticF:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-cajgfk
Out[1]=1

Integral representation:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-bp7jmj
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-eu1gey

Applications  (4)Sample problems that can be solved with this function

Solution of the pendulum equation in the overswing mode:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-izefg0

Check with the pendulum equation:

In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-c92izb
Out[2]=2

Plot the solution:

In[3]:=3
✖
https://wolfram.com/xid/0dc3l0sh425c-bahnfv
Out[3]=3

Relativistic solution of the sine‐Gordon equation:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-g2iof6

Verify the solution by substituting into the sine‐Gordon equation:

In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-h9yil3
Out[2]=2

Plot the solution for different values of and :

In[3]:=3
✖
https://wolfram.com/xid/0dc3l0sh425c-h6ibmm
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0dc3l0sh425c-jnopnz
Out[4]=4

Form and plot generalized Fourier series:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-p8r6y
Out[1]=1

Spherical triangle from an elliptic parameter and triple such that :

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-cvpgnt
In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-c19p3g

Angles at vertices , and and sides opposite to each corresponding vertex:

In[3]:=3
✖
https://wolfram.com/xid/0dc3l0sh425c-lvlsl5
Out[3]=3

Verify Cagnoli's equation, which relates the measurements of all the angles and sides of a spherical triangle:

In[4]:=4
✖
https://wolfram.com/xid/0dc3l0sh425c-ig9e5d
Out[4]=4

Compute the area of the triangle, also known as the spherical excess:

In[5]:=5
✖
https://wolfram.com/xid/0dc3l0sh425c-hjye70
Out[5]=5

Compute the excess using L'Huilier's theorem [MathWorld]:

In[6]:=6
✖
https://wolfram.com/xid/0dc3l0sh425c-dyaagc
Out[6]=6

Compute corresponding points on 3D unit sphere:

In[7]:=7
✖
https://wolfram.com/xid/0dc3l0sh425c-ibp9i3

Check consistency between points found and spherical sides:

In[8]:=8
✖
https://wolfram.com/xid/0dc3l0sh425c-b32d38
Out[8]=8

Visualize the triangle:

In[9]:=9
✖
https://wolfram.com/xid/0dc3l0sh425c-ky6u8o
Out[9]=9

Properties & Relations  (5)Properties of the function, and connections to other functions

Compose with inverse functions:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-emofur
Out[1]=1

Use PowerExpand to disregard the multivaluedness of the inverse function:

In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-fzgyvy
Out[2]=2

Apply trigonometric functions to JacobiAmplitude:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-kmykn1
Out[1]=1

Solve a transcendental equation:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-iy4yu
Out[1]=1

Obtain from a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-ntn94
Out[1]=1

JacobiAmplitude has branch cuts from 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+1) TemplateBox[{{1, -, m}}, EllipticK] to 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+3) TemplateBox[{{1, -, m}}, EllipticK] for any integers and :

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-gap67e
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-c8jrqk
Out[2]=2

Visualize branch cuts for different moduli:

In[3]:=3
✖
https://wolfram.com/xid/0dc3l0sh425c-ezy01e
In[4]:=4
✖
https://wolfram.com/xid/0dc3l0sh425c-drxec9
Out[4]=4

Possible Issues  (1)Common pitfalls and unexpected behavior

The branch cuts of TemplateBox[{u, m}, JacobiAmplitude] for cross the real axis at 2(2 s+1) TemplateBox[{{1, /, m}}, EllipticK]/sqrt(m) for integer :

In[1]:=1
✖
https://wolfram.com/xid/0dc3l0sh425c-78udx
Out[1]=1

For physical applications where a continuous version of the amplitude is desired for , use the following definition:

In[2]:=2
✖
https://wolfram.com/xid/0dc3l0sh425c-fgte3b

The preceding function coincides with sin^(-1)(TemplateBox[{u, m}, JacobiSN]) for real and :

In[3]:=3
✖
https://wolfram.com/xid/0dc3l0sh425c-kpz54
Out[3]=3
Wolfram Research (1988), JacobiAmplitude, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiAmplitude.html (updated 2020).
✖
Wolfram Research (1988), JacobiAmplitude, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiAmplitude.html (updated 2020).

Text

Wolfram Research (1988), JacobiAmplitude, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiAmplitude.html (updated 2020).

✖
Wolfram Research (1988), JacobiAmplitude, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiAmplitude.html (updated 2020).

CMS

Wolfram Language. 1988. "JacobiAmplitude." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/JacobiAmplitude.html.

✖
Wolfram Language. 1988. "JacobiAmplitude." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/JacobiAmplitude.html.

APA

Wolfram Language. (1988). JacobiAmplitude. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiAmplitude.html

✖
Wolfram Language. (1988). JacobiAmplitude. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiAmplitude.html

BibTeX

@misc{reference.wolfram_2025_jacobiamplitude, author="Wolfram Research", title="{JacobiAmplitude}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiAmplitude.html}", note=[Accessed: 26-April-2025 ]}

✖
@misc{reference.wolfram_2025_jacobiamplitude, author="Wolfram Research", title="{JacobiAmplitude}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiAmplitude.html}", note=[Accessed: 26-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_jacobiamplitude, organization={Wolfram Research}, title={JacobiAmplitude}, year={2020}, url={https://reference.wolfram.com/language/ref/JacobiAmplitude.html}, note=[Accessed: 26-April-2025 ]}

✖
@online{reference.wolfram_2025_jacobiamplitude, organization={Wolfram Research}, title={JacobiAmplitude}, year={2020}, url={https://reference.wolfram.com/language/ref/JacobiAmplitude.html}, note=[Accessed: 26-April-2025 ]}