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Complete elliptic integrals of the first and second kinds

`[`

returns the complete
elliptic integrals of the first and second kinds.`K`

,`E`

] =
ellipke(`m`

)

Compute the complete elliptic integrals of the first and second kinds for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

[K0, E0] = ellipke(0) [K05, E05] = ellipke(1/2)

K0 = 1.5708 E0 = 1.5708 K05 = 1.8541 E05 = 1.3506

Compute the complete elliptic integrals for the same numbers
converted to symbolic objects. For most symbolic (exact) numbers, `ellipke`

returns
results using the `ellipticK`

and `ellipticE`

functions.

[K0, E0] = ellipke(sym(0)) [K05, E05] = ellipke(sym(1/2))

K0 = pi/2 E0 = pi/2 K05 = ellipticK(1/2) E05 = ellipticE(1/2)

Use `vpa`

to approximate `K05`

and `E05`

with
floating-point numbers:

vpa([K05, E05], 10)

ans = [ 1.854074677, 1.350643881]

`0`

and `1`

If the argument does not belong to the range from 0 to 1, then
convert that argument to a symbolic object before using `ellipke`

:

[K, E] = ellipke(sym(pi/2))

K = ellipticK(pi/2) E = ellipticE(pi/2)

Alternatively, use `ellipticK`

and `ellipticE`

to
compute the integrals of the first and the second kinds separately:

K = ellipticK(sym(pi/2)) E = ellipticE(sym(pi/2))

K = ellipticK(pi/2) E = ellipticE(pi/2)

Call `ellipke`

for this symbolic matrix. When
the input argument is a matrix, `ellipke`

computes
the complete elliptic integrals of the first and second kinds for
each element.

[K, E] = ellipke(sym([-1 0; 1/2 1]))

K = [ ellipticK(-1), pi/2] [ ellipticK(1/2), Inf] E = [ ellipticE(-1), pi/2] [ ellipticE(1/2), 1]

Calling

`ellipke`

for numbers that are not symbolic objects invokes the MATLAB^{®}`ellipke`

function. This function accepts only`0 <= m <= 1`

. To compute the complete elliptic integrals of the first and second kinds for the values out of this range, use`sym`

to convert the numbers to symbolic objects, and then call`ellipke`

for those symbolic objects. Alternatively, use the`ellipticK`

and`ellipticE`

functions to compute the integrals separately.For most symbolic (exact) numbers,

`ellipke`

returns results using the`ellipticK`

and`ellipticE`

functions. You can approximate such results with floating-point numbers using`vpa`

.If

`m`

is a vector or a matrix, then`[K,E] = ellipke(m)`

returns the complete elliptic integrals of the first and second kinds, evaluated for each element of`m`

.

You can use `ellipticK`

and `ellipticE`

to compute elliptic integrals
of the first and second kinds separately.

[1] Milne-Thomson, L. M. “Elliptic Integrals.” *Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical
Tables.* (M. Abramowitz and I. A. Stegun, eds.). New York:
Dover, 1972.