       RE: implicit function

• To: mathgroup at smc.vnet.net
• Subject: [mg24238] RE: [mg24218] implicit function
• From: "David Park" <djmp at earthlink.net>
• Date: Mon, 3 Jul 2000 20:39:19 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```
> -----Original Message-----
> From: sang jo lee [mailto:sjlee at physics.yeungnam.ac.kr]
To: mathgroup at smc.vnet.net

> I'm a graduate student in South Korea for Physics and my name is Sangjo
> Lee.
>
> Following is my problem.
> there is one implicit function like this :
> ( (1-y) / x + 1/m )^x = y Exp[f(x) z]  / m^x     where     f(x) = 1 + (x
> - 1)/( 2( x + 1) )
> x, y is variable and other symbols are constants.
> I want to get the numerical y value when x is  -10 < x < 10.
>
> well...
> I am looking forward to seeing your e-mail.
> Thank you.
>
> yours Sincerely.
>
> from  S. Korea
>

Sanjo,

It would help if we knew what values of m and z you were interested in. Are
you looking for complex solutions or only real solutions?

If you are looking only for real solutions, then the following code will
pick off many of them.

f[x_] := 1 + (x - 1)/(2*(x + 1))
eqn[m_, z_] := ((1 - y)/x + 1/m)^x == y*Exp[f[x]*z]/m^x

This works with the Log of the equation.

Simplify[eqn[m, z]]
Log /@ %
MapAt[Together, %, {1, 1, 1}]
PowerExpand[%]
logeqn[m_, z_][x_, y_] = Simplify[%]

which gives the final result:

(z + 3*x*z)/(-2 - 2*x) + x*Log[m + x - m*y] == x*Log[x] + Log[y]

To keep all values real (but would it be possible to obtain real solutions
with complex intermediate results?) we must have x > 0, y > 0 and y <
(x+m)/m for m positive. Then the following code can be used to solve for
most y values for positive x.

With[
{m = 2,
z = 3,
delta = 0.0001},
FindRoot[Evaluate[logeqn[m, z][#, y]],
Evaluate[{y, {delta, (# + m)/m - delta}}]] & /@ {0.1, 1, 2, 3, 4, 5,
6, 7, 8, 9, 10}]
{{y -> 0.22483877308760597}, {y -> 0.13583550274287187}, {y ->
0.10808659209094279},
{y -> 0.09672372500216456}, {y -> 0.09064027759035838},
{y -> 0.08687290279481108}, {y -> 0.0843176048864707}, {y ->
0.08247327685930991},
{y -> 0.08108067038556814}, {y -> 0.0799925440751478}, {y ->
0.07911920499390443}}

David Park