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e^x=x


jpat1023

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The problem is to find the real zeros of a nonlinear function f(x)=0.

 

Let [math]y = e^x[/math] and [math]y = x[/math]

 

Hence [math]e^x = x[/math]

 

[math]e^x - x = 0[/math]

 

Since the two function do not intersect there is no solution

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The problem is to find the real zeros of a nonlinear function f(x)=0.

 

Let [math]y = e^x[/math] and [math]y = x[/math]

 

Hence [math]e^x = x[/math]

 

[math]e^x - x = 0[/math]

 

Since the two function do not intersect there is no solution

 

buts thats only for [math]x \in \mathbb{R}[/math] . there might still be a solution where [math]x \in \mathbb{C}[/math]

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For the case when [math] x \in \mathbb{C} [/math]

 

Using two terms of the taylor series expansion we have

 

[math] e^x - x = 0 [/math]

 

[math] 1 + x + \frac {x^2}{2} - x = 0 [/math]

 

[math] 1 + \frac {x^2}{2} = 0 [/math]

 

[math]x_1 = (0 + i\sqrt {2})[/math]

 

[math]x_2 = (0 - i\sqrt {2})[/math]

 

Using three terms of the taylor series expansion we have

 

[math] 1 + x + \frac {x^2}{2} + \frac {x^3}{6} - x = 0 [/math]

 

[math]x_1 = (0.246017 + i1.28751)[/math]

 

[math]x_2 = (0.246017 - i1.28751)[/math]

 

[math]x_3 = -3.49203[/math]

 

The number of solutions you require depends on how many terms you use in the taylor series expansion.

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i tried to use MAPLE to solve it. and it gave me this

 

the command was

 

solve(e^x=x,x); which basically says solve e^x=x for x

 

and the output was.

 

[math]\frac{-LambertW(-ln(e))}{ln(e)}[/math]

 

dont know why it doesnt automatically simplify

 

so its just [math]$LambertW$[/math](-1)

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evalf think uses floating point arithmetic . tried that. seems to give error

 

tried evalc which seemed to be the more correct command. and got some long giberrish with some functions i havent even seen before

 

Isnt the whole point of what you are trying to do : approximating the lambertW function with some complex number ?

Analytical expressions would simply give x = -LambertW(-1) as a solution.

 

have you tried using command line maple also ? evalf(-LambertW(-1)); ?

 

Mandrake

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