## RJ FunctionMath Engineering Class

```Public Function RJ( _
ByVal vX As Variant _
, ByVal vY As Variant _
, ByVal vZ As Variant _
, ByVal vP As Variant _
) As Variant```

### Compute the incomplete or complete (X or Y or Z is zero) elliptic integral of the 3rd kind.

Examples:
```    RJ(1, 1, 3, 4) = 0.286898213878455
RJ(1, 2, 3, 4) = 0.239848099749568
RJ(2, 1, 3, 4) = 0.239848099749568
RJ(2, 2, 3, 4) = 0.202320259296164```
```    RC Function
RD Function
RF Function
DRJ Function```
vX: Function returns Null if vX is Null or cannot be fixed up to a Double precision floating point number.
vY: Function returns Null if vY is Null or cannot be fixed up to a Double precision floating point number.
vZ: Function returns Null if vZ is Null or cannot be fixed up to a Double precision floating point number.
vP: Function returns Null if vP is Null or cannot be fixed up to a Double precision floating point number.

For X, Y, and Z non-negative, at most one of them zero, and P positive, RJ(X,Y,Z,P) = Integral from zero to infinity of (3/2)(t+X)^(-1/2)(t+Y)^(-1/2)(t+Z)^(-1/2)(t+P)^(-1) dt.

The routine calculates an approximation result to RJ(X,Y,Z,P) = Integral from zero to infinity of

```              -1/2     -1/2     -1/2     -1
(3/2)(t+X)    (t+Y)    (t+Z)    (t+P)  dt,```
where X, Y, and Z are nonnegative, at most one of them is zero, and P is positive. If X or Y or Z is zero, the integral is COMPLETE. The duplication theorem is iterated until the variables are nearly equal, and the function is then expanded in Taylor series to fifth order.