Does this equation have any solution?

  • Thread starter Petrushka
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In summary, the conversation discusses the existence of complex solutions to the equation x\multsp =\multsp {{e}^x}, with the conclusion that there are two distinct solutions given by x_{1}=-W(-1) and x_{2}=-W(-1,-1), where W is the Lambert function. The conversation also touches on the confusion and clarification surrounding the nature of x_{1}=-W(-1) as a complex number and the lack of real solutions for the equation.
  • #1
Petrushka
18
0
[tex]x\multsp =\multsp {{e}^x}[/tex]

I'm aware there's no real solution, but does any complex solution exist?
 
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  • #2
Yes,it has 2 distinct solutions:

[tex] x_{1}=-W(-1) [/tex]

[tex] x_{2}=-W(-1,-1) [/tex]

,where W is the Lambert function.

Daniel.
 
  • #3
I see. Having read a bit about the Lambert function on Http://www.mathworld.com it doesn't make much sense to me.

Is the following value:

[tex]x_{1}=-W(-1)[/tex]

...a complex number, or something entirely different?
 
  • #4
You mean a # of nonzero imaginary part...?Yes it is.As u said and as a graph and elementary calculus would show,for real #-s the equation does not have solution.

Daniel.
 

1. How do I know if an equation has a solution?

There are a few ways to determine if an equation has a solution. One method is to graph the equation and see if it intersects with the x-axis. If it does, then the equation has a solution. Another method is to solve the equation algebraically and see if the solution is a real number. If it is, then the equation has a solution.

2. Can an equation have more than one solution?

Yes, an equation can have more than one solution. This is often the case with quadratic equations, which can have two solutions. However, some equations may have an infinite number of solutions, such as linear equations with the form y=x. It all depends on the type of equation and the values of the variables involved.

3. What does it mean if an equation has no solution?

If an equation has no solution, it means that there is no value for the variable that satisfies the equation. In other words, there is no number that you can plug in for the variable that will make the equation true. This can happen when the equation is contradictory, such as 2x=3 and x=4, as there is no number that can satisfy both equations simultaneously.

4. Can an equation have complex solutions?

Yes, an equation can have complex solutions. Complex solutions involve the use of imaginary numbers, which are numbers that involve the square root of -1. These types of solutions often occur in quadratic equations, and they are denoted by the letter "i" in the solution. For example, the solution to the equation x^2+1=0 is x= ±i.

5. How can I check if my solution to an equation is correct?

To check if your solution to an equation is correct, you can plug the solution back into the original equation and see if it makes the equation true. If it does, then your solution is correct. You can also graph the equation and see if the solution falls on the point where the equation intersects with the x-axis. Additionally, you can use a calculator or online tool to solve the equation and compare your solution to the one given by the tool.

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