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EllipticK
✖
EllipticK

✖

gives the complete elliptic integral of the first kind TemplateBox[{m}, EllipticK].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • EllipticK is given in terms of the incomplete elliptic integral of the first kind by TemplateBox[{m}, EllipticK]=TemplateBox[{{pi, /, 2}, m}, EllipticF].
  • EllipticK[m] has a branch cut discontinuity in the complex m plane running from to .
  • For certain special arguments, EllipticK automatically evaluates to exact values.
  • EllipticK can be evaluated to arbitrary numerical precision.
  • EllipticK automatically threads over lists.
  • EllipticK can be used with Interval and CenteredInterval objects. »

Examples

open allclose all

Basic Examples  (5)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-fy1s9a
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-el2ern
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-kiedlx
Out[1]=1

Series expansion at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-crmtmd
Out[1]=1

Series expansion at Infinity:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-laddhh
Out[1]=1

Scope  (38)Survey of the scope of standard use cases

Numerical Evaluation  (5)

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-jg2h28
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-cvhbc3
Out[2]=2

Evaluate numerically for complex arguments:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-yk2
Out[1]=1

Evaluate EllipticK efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-di5gcr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-bq2c6r
Out[2]=2

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-gm995c
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-cmdnbi
Out[2]=2

Or compute average-case statistical intervals using Around:

In[3]:=3
✖
https://wolfram.com/xid/0e7ooerm-cw18bq
Out[3]=3

Compute the elementwise values of an array:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-thgd2
Out[1]=1

Or compute the matrix EllipticK function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-o5jpo
Out[2]=2

Specific Values  (5)

Simple exact values are generated automatically:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-cfuygx
Out[1]=1

Some exact values in terms of Gamma after applying FunctionExpand:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-hzrfoj
Out[1]=1

Find directional limiting values at branch cuts:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-iasiu4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-okkyj3
Out[2]=2

Value at infinity:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-bdij6w
Out[1]=1

Find the root of the equation TemplateBox[{m}, EllipticK]=2:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-f2hrld
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-jegwkj
Out[2]=2

Visualization  (2)

Plot EllipticK:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-ecj8m7
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-lmzfrf
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-co5mrt
Out[2]=2

Function Properties  (9)

EllipticK is defined for all real values less than 1:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-cl7ele
Out[1]=1

EllipticK takes all real positive values:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-evf2yr
Out[1]=1

EllipticK is not an analytic function:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-h5x4l2
Out[1]=1

Has both singularities and discontinuities:

In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-mdtl3h
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e7ooerm-mn5jws
Out[3]=3

EllipticK is not a meromorphic function:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-l9hj76
Out[1]=1

EllipticK is nondecreasing on its domain:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-nlz7s
Out[1]=1

EllipticK is injective:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-poz8g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-ctca0g
Out[2]=2

EllipticK is not surjective:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-cxk3a6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-frlnsr
Out[2]=2

EllipticK is non-negative on its domain:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-84dui
Out[1]=1

EllipticK is convex on its domain:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-8kku21
Out[1]=1

Differentiation  (3)

First derivative:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-mmas49
Out[1]=1

Higher derivatives:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-nfbe0l
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-fuzyd6
Out[2]=2

Formula for the ^(th) derivative:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-nig1uv
Out[1]=1

Integration  (3)

Indefinite integral of EllipticK:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-bponid
Out[1]=1

Definite integral over an interval lying on the branch cut:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-bo0zm6
Out[1]=1

More integrals:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-jcfk61
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-muvmkf
Out[2]=2

Series Expansions  (3)

Taylor expansion for EllipticK:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-ewr1h8
Out[1]=1

Plot the first three approximations for EllipticK around :

In[7]:=7
✖
https://wolfram.com/xid/0e7ooerm-binhar
Out[8]=8

Series expansions at branch points:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-b62ffq
Out[1]=1

EllipticK can be applied to power series:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-e4s3cf
Out[1]=1

Integral Transforms  (3)

Compute the Laplace transform using LaplaceTransform:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-d7r6v4
Out[1]=1

MellinTransform:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-eqbky1
Out[1]=1

HankelTransform:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-7mn4u
Out[1]=1

Function Representations  (5)

Relation to other elliptic integrals:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-dy82qg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-cml0qj
Out[2]=2

Relation to the LegendreP:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-k8bx14
Out[1]=1

Represent in terms of MeijerG using MeijerGReduce:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-e1b67p
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-b33lgc
Out[2]=2

EllipticK can be represented as a DifferentialRoot:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-bgjnbg
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-k24z50

Applications  (7)Sample problems that can be solved with this function

Small-angle approximation to the period of a pendulum:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-bveji5
Out[1]=1

Plot the period versus the initial angle:

In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-erjyel
Out[2]=2

Vector potential due to a circular current flow, in cylindrical coordinates:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-zoxoa

The components of the magnetic field:

In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-bu2wd7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e7ooerm-elt4c6
Out[3]=3

Plot the magnitude of the magnetic field:

In[4]:=4
✖
https://wolfram.com/xid/0e7ooerm-qcw24
Out[4]=4

Resistance between the origin and the point in an infinite 3D lattice of unit resistors:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-jd7ddr
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-irxcjg
Out[2]=2

Energy for the Onsager solution of the Ising model:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-ihgp9x

Plot of the specific heat:

In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-cspo98
Out[2]=2

Find the critical temperature:

In[3]:=3
✖
https://wolfram.com/xid/0e7ooerm-eacjy
Out[3]=3

Calculate a singular value:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-8jprv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-hgrz7
Out[2]=2

Current flow in a rectangular conducting sheet with voltage applied at a pair of opposite corners:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-vc9tw

Plot the flow lines with bounds defined via EllipticK:

In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-rxs56
Out[2]=2

Construct lowpass elliptic filter for analog signal:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-bxw8jc

Compute filter ripple parameters and associate elliptic function parameter:

In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-bm1kjg

Use elliptic degree equation to find the ratio of the pass and the stop frequencies:

In[3]:=3
✖
https://wolfram.com/xid/0e7ooerm-b13u0f
Out[3]=3

Compute corresponding stop frequency and elliptic parameters:

In[4]:=4
✖
https://wolfram.com/xid/0e7ooerm-wtjxg
Out[4]=4

Compute ripple locations and poles and zeros of the transfer function:

In[5]:=5
✖
https://wolfram.com/xid/0e7ooerm-kvrem

Compute poles of the transfer function:

In[6]:=6
✖
https://wolfram.com/xid/0e7ooerm-elkxl1

Assemble the transfer function:

In[7]:=7
✖
https://wolfram.com/xid/0e7ooerm-fyal2v
In[8]:=8
✖
https://wolfram.com/xid/0e7ooerm-bv4lrk
Out[8]=8

Compare with the result of EllipticFilterModel:

In[9]:=9
✖
https://wolfram.com/xid/0e7ooerm-00j7s
In[10]:=10
✖
https://wolfram.com/xid/0e7ooerm-ef2b35
Out[10]=10

Properties & Relations  (4)Properties of the function, and connections to other functions

This shows the branch cuts of the EllipticK function:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-guz5j
Out[1]=1

Numerically find a root of a transcendental equation:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-7qwnc
Out[1]=1

Solve a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-cidt66
Out[1]=1

EllipticK is a particular case of various mathematical functions:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-csy6db
Out[1]=1

Possible Issues  (3)Common pitfalls and unexpected behavior

Machine-precision evaluation can result in numerically inaccurate answers near branch cuts:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-llt4w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-bnko5x
Out[2]=2

The defining integral converges only under additional conditions:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-b7kr8t
Out[1]=1

Different argument conventions exist that result in modified results:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-eecbzs
Out[1]=1

Neat Examples  (2)Surprising or curious use cases

Probability that a random walker in a 3D cubic lattice returns to the origin:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-es3h5p
Out[1]=1

Carry out a modeling run of 1000 walks and count how many it returns to the origin:

In[2]:=2
✖
https://wolfram.com/xid/0e7ooerm-b3nl4n
Out[3]=3

Compare with the expected count at :

In[4]:=4
✖
https://wolfram.com/xid/0e7ooerm-berj2o
Out[4]=4

Riemann surface of TemplateBox[{m}, EllipticK]:

In[1]:=1
✖
https://wolfram.com/xid/0e7ooerm-nujfsc
Out[1]=1
Wolfram Research (1988), EllipticK, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticK.html (updated 2022).
✖
Wolfram Research (1988), EllipticK, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticK.html (updated 2022).

Text

Wolfram Research (1988), EllipticK, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticK.html (updated 2022).

✖
Wolfram Research (1988), EllipticK, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticK.html (updated 2022).

CMS

Wolfram Language. 1988. "EllipticK." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/EllipticK.html.

✖
Wolfram Language. 1988. "EllipticK." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/EllipticK.html.

APA

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

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

BibTeX

@misc{reference.wolfram_2025_elliptick, author="Wolfram Research", title="{EllipticK}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticK.html}", note=[Accessed: 26-March-2025 ]}

✖
@misc{reference.wolfram_2025_elliptick, author="Wolfram Research", title="{EllipticK}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticK.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_elliptick, organization={Wolfram Research}, title={EllipticK}, year={2022}, url={https://reference.wolfram.com/language/ref/EllipticK.html}, note=[Accessed: 26-March-2025 ]}

✖
@online{reference.wolfram_2025_elliptick, organization={Wolfram Research}, title={EllipticK}, year={2022}, url={https://reference.wolfram.com/language/ref/EllipticK.html}, note=[Accessed: 26-March-2025 ]}