Finding Two-Term Asymptotic Expansion for Real Roots of xe-x=epsilon

  • Thread starter Thread starter haywood
  • Start date Start date
AI Thread Summary
To find a two-term asymptotic expansion for the real roots of the equation xe^(-x) = epsilon as epsilon approaches 0, a Taylor polynomial can be used for the root near 0. For the second root, taking logarithms of both sides is an effective approach. This method provides a clearer pathway to approximate the roots. The discussion highlights the importance of these techniques in deriving asymptotic expansions. Overall, the conversation focuses on practical methods for solving the equation.
haywood
Messages
4
Reaction score
0
Can anyone tell me how to find a two-term asymptotic expansion for the two real roots of xe-x=epsilon as epsilon --> 0

Thanks,
A.Haywood
 
Mathematics news on Phys.org
dear Haywood
I don't know what kind of asypmtotic expasion you are looking for, so a short hint on how to solve the equation approximatly must suffice.
For the root near 0 substitute the left hand side with a taylor polynomial.
For the other root take logarithms on both sides.
 
That works! I was especially looking for how to find that second root, so grateful that you mentioned using ln :-)
Thank you, Dalle!
 

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »
Thread 'Imaginary Pythagorus'

Thread 'Imaginary Pythagorus'

I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
View full post »

Similar threads

How should I show that this root is given approximately by this?
Replies
2
Views
2K
MHB Real Roots of Composite Polynomials: Solving P(Q(x))=0
Replies
1
Views
1K
I Make Intelligent Initial Guesses for Newton-Raphson Programmatically
Replies
16
Views
3K
I Quartic real roots - factor part into quadratic
Replies
3
Views
1K
I Explicit logical justification for last step in epsilon/delta proof?
Replies
20
Views
1K
MHB How can we determine the sum of roots in a quadratic equation with real roots?
Replies
4
Views
2K
MHB Can All Roots of a Quartic Polynomial Be Real?
Replies
1
Views
1K
MHB Finding $b(a+c)$ for Real Roots of $\sqrt{2014}x^3-4029x^2+2=0$
Replies
1
Views
1K
MHB Finding Min Value of $\dfrac{|b|+|c|}{a}$ from Roots of Cubic Equations
Replies
4
Views
1K
B Principal square root of a complex number, why is it unique?
Replies
13
Views
5K

Hot Threads

Recent Insights

Back
Top