Solve Simple Residue Question with Laplace Transforms | Limit as p->0

  • Thread starter Thread starter TDJHOLDER
  • Start date Start date
  • Tags Tags
    Residue
Click For Summary
SUMMARY

The discussion focuses on calculating the residue of the function (1/p)((sinh(p^1/2)(1-x))/(sinh(p^1/2))) as p approaches 0. The user is attempting to find the limit after multiplying by p, which leads to an indeterminate form. The conclusion drawn is that the residue at p=0 is indeed (1-x), despite the complications arising from the hyperbolic sine terms in the denominator.

PREREQUISITES
  • Understanding of Laplace transforms
  • Familiarity with hyperbolic functions, specifically sinh
  • Knowledge of residue theory in complex analysis
  • Experience with limits and indeterminate forms
NEXT STEPS
  • Study the properties of Laplace transforms in detail
  • Learn about residue calculation techniques in complex analysis
  • Explore the behavior of hyperbolic functions near zero
  • Practice solving limits involving indeterminate forms
USEFUL FOR

Students and professionals in mathematics, particularly those focusing on complex analysis and Laplace transforms, as well as anyone interested in advanced calculus techniques.

TDJHOLDER
Messages
2
Reaction score
0
Hi guys,

I'm doing some Laplace transforms
Apologies for not typing this out very well I don't know LaTex...
where (1/p)((sinh(p^1/2)(1-x))/(sinh(p^1/2)))

I need to work out the residue at p=0...

It's been a while since I did this but you multiply through by p and then take the limit as p->0

I can't seem to work the last bit out as the bottom of the fraction becomes zero.

I'm guessing the residue is probably (1-x) but I don't know how to get there.

Thanks in advance.
 
Physics news on Phys.org
Don't the hyperbolic sines just cancel?
 

Similar threads

Mellin transform of Dirac delta function ##\delta(t-a)##
  • · Replies 3 ·
Replies
3
Views
3K
Determine whether ## S[y] ## has a maximum or a minimum
  • · Replies 18 ·
Replies
18
Views
3K
A tricky inverse Laplace transform
  • · Replies 8 ·
Replies
8
Views
3K
Laplace transform of cosine squared function
  • · Replies 1 ·
Replies
1
Views
1K
If a and b are both quadratic residues/nonresidues mod p & q
  • · Replies 1 ·
Replies
1
Views
2K
What is the Inverse Laplace Transform of e^(-sx^2/2)?
  • · Replies 10 ·
Replies
10
Views
3K
Random number generation: probability of 2nd smallest >0.002
  • · Replies 8 ·
Replies
8
Views
1K
Partial fraction decomposition with Laplace transformation in ODE
  • · Replies 9 ·
Replies
9
Views
3K
Elliptic functions, residue computation, same zeros and poles
  • · Replies 1 ·
Replies
1
Views
2K
Solving ##y'' - 5 y' - 6y = e^{3x}## using Laplace Transform
  • · Replies 7 ·
Replies
7
Views
2K