# Question:elliptic integral

## Question:elliptic integral

Maple

Hi all,

While solving a différential equation, I encounter an integral:

```> Int(1/sqrt(1-k^2*sin(s)^2), s = 0 .. phi);

/phi
|               1
|     --------------------- ds
|                     (1/2)
/0     /     2       2\
\1 - k  sin(s) /

```

with k<1.  But Maple don't want to give me the special function F(k,phi)

```>value(%) assuming k < 1;

/          1                        \
int|---------------------, s = 0 .. phi|
|                (1/2)              |
|/     2       2\                   |
\\1 - k  sin(s) /                   /

```

What I found in Abramowitz, is that if phi vary from 0 to Pi/2, it's an elliptic integral of the first kind. (p. 608)

```> Int(1/sqrt(1-k^2*sin(s)^2), s = 0 .. (1/2)*Pi);

/1/2 Pi
|                  1
|        --------------------- ds
|                        (1/2)
/0        /     2       2\
\1 - k  sin(s) /
> value(%) assuming 0 < k  < 1;

/   1\
EllipticF|k, -|
\   k/
---------------
k

```

but I do need from 0 to phi.

In addition, there is a reference about the Jacobi elliptic function (p.569) wich is suppose to be the inverse of the integral I want to solve.

Can someone untangle this for me.

Thanks in advance

Mario

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