HP49-50G,VER16.2 Lambert Funktion Wk(x), k=k= 0, ±1, ±2, ±3, ±4..., x real or complex

[Gil]

New version, 8.02, with my thanks to Albert Chan.

The program accepts now all kind of argument : real(x) or imaginary (x) —>± infinity, for any integer branch k.

Input {k x}, x complex number or not.

Let's try some large values of x (large for the real or for the imaginary part)

1) x=1E450, k=0

Input in HP50G
1E450

or
{0, 1E450}

Then press
LAMBERT

Output
:W0
(
1.E450
)
: 1029.2267288

Does it go to +inf?
Try with Wolfram
Write LambertW (0,1E5000000)
Output: 1.15E7

Or write LambertW (0, infinity)
Output: infinity

And with HP50G?
Write sign for infinity
LS+0
Then press LAMBERT
Output: 'infinity'

2) x=1E450, k=15

Input in HP50G
{15 1E450}

Then press
LAMBERT

Output

:W15
(
1.E450
)
: (1029.22256567,94.1565503694)

Does that result go to +inf (with no imaginary part, here = 94,16)
Try with Wolfram
Write LambertW (15,1E5000000)
Output: (1.15E7+94.24777)
And you see that 94.24777=30pi

But, if you try and write LambertW (0, infinity)
Output: infinity.
Well, let's admit it is correct.
Next example will put in question that kind of answer.

And with HP50G?
Write {15 infinity}
(LS+0 for infinty)
Then press LAMBERT
Output: "infinity + i(30pi)"


3) x=(-1E450), k=0,

Input in HP50G
(0,-1E450)

or
{0, - 1E450}

Then press
LAMBERT

Output
{ "W0
(
-inf
)
:" inf-ipi" }

Does it go to +inf?
Try with Wolfram
Write LambertW(-9999E5000000)
Output: 1.15E7+i*3.14159238 (almost pi)

Or write LambertW (-i*infinity)
Output: infinity?!

And with HP50G?
Write sign for infinity
LS+0
Then press key ± (letter M)
Then press LAMBERT
Output: "infinity+ i(pi)"


4) x=(0, 1E450), k=0,
ie a large imaginary part

Input in HP50G
(0,1E450)

or
{0, (0,1E450)}

Then press
LAMBERT

Output
::W0
(
(0.,1.E450)
)
: (1029.22672764,1.56927161862)

Does it go to +inf?
Try with Wolfram
Write LambertW(0+i*9999E5000000)
Output: 1.15E7+i*1.570796

Or write LambertW (0+i*infinity)
Output: infinity?!

And with HP50G?
Write sign for infinity
LS+0
Then press LAMBERT
Output: "infinity+ i(pi/2)"

5) case x=(inf-i*inf), k= 30
Write 30
ENTER
key oo, ie LS+0[/code]
—>NUM
DUP
key ± (letter M)
R—>C (to get together inf, real part, and -inf, imaginary part, into a complex number)
2 —> list
You should get the following list
{30, (9.999999999E499, 9.999999999E499)}
Then press LAMBERT on the HP50G.

Output on HP50G
"oo +i(60pi).

Approximation with Wolfram
LambertW(30, 1E10000+i*E10000)
Result: 23015.807+i×188.487, about i×(60 pi)[code]