Dirac delta integrated from 0 to infinity

Click For Summary
SUMMARY

The integration of the Dirac delta function from 0 to infinity yields different results based on the direction of approach to 0. Specifically, using the limit as \( a \) approaches 0 from the left ( \( \lim_{a\rightarrow{0^-}} \int_a^\infty \delta(x) dx \) ) results in a value of 1, while approaching from the right ( \( \lim_{a\rightarrow{0^+}} \int_a^\infty \delta(x) dx \) ) results in 0. The Dirac delta function is symmetric, leading to a conclusion that the integral can be interpreted as yielding 0.5 when considering both limits. This discussion clarifies the nuances of integrating the Dirac delta function in different contexts.

PREREQUISITES
  • Understanding of the Dirac delta function
  • Familiarity with limits in calculus
  • Knowledge of integral calculus
  • Concept of symmetric functions
NEXT STEPS
  • Study the properties of the Dirac delta function in detail
  • Learn about limits and their applications in calculus
  • Explore symmetric functions and their implications in integration
  • Investigate advanced topics in distribution theory
USEFUL FOR

Students, mathematicians, and physicists interested in advanced calculus, particularly those working with distributions and the Dirac delta function.

Heimisson
Messages
42
Reaction score
0
I was wondering if I integrate the dirac delta function from 0 to infinity where the function it's integrated with is the constant 1, will I get 0.5 or 1? And why?

This is not homework so I decided to post this here although I asked this question in class and the teacher (assistant) wasn't too sure. I could ask my professor but he's a scary man and mocks your questions.

So I was hoping someone here could answer this.

thanks
 
Physics news on Phys.org
Heimisson said:
I was wondering if I integrate the dirac delta function from 0 to infinity where the function it's integrated with is the constant 1, will I get 0.5 or 1? And why?

This is not homework so I decided to post this here although I asked this question in class and the teacher (assistant) wasn't too sure. I could ask my professor but he's a scary man and mocks your questions.

So I was hoping someone here could answer this.

thanks

Depends on which direction you approach 0 from. If your integral is...

lim_{a\rightarrow{0^-}} \int_a^\infty \delta(x) dx

Then the answer is 1. If your integral is...


lim_{a\rightarrow{0^+}} \int_a^\infty \delta(x) dx

Then the answer is 0.
 
You get 1/2, Dirac delta is a symmetric function.
 
peteratcam said:
You get 1/2, Dirac delta is a symmetric function.

It's not even really a function...
 
Char. Limit said:
Depends on which direction you approach 0 from. If your integral is...

lim_{a\rightarrow{0^-}} \int_a^\infty \delta(x) dx

Then the answer is 1. If your integral is...


lim_{a\rightarrow{0^+}} \int_a^\infty \delta(x) dx

Then the answer is 0.

Thanks a lot this makes sense.
 

Similar threads

Undergrad Matrix elements of Hamiltonian include Dirac delta term
  • · Replies 2 ·
Replies
2
Views
1K
Graduate Going from Kronecker deltas to Dirac deltas
  • · Replies 15 ·
Replies
15
Views
2K
Graduate A particle locked inside an arrangement of Dirac delta potentials
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad Questions about the Dirac delta
  • · Replies 26 ·
Replies
26
Views
5K
Graduate Delta function and dirac notation
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Show that the integral of the Dirac delta function is equal to 1
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Dirac-Delta Function, Different Integration Variable
  • · Replies 14 ·
Replies
14
Views
3K
Graduate Dirac Delta: Normal -> Lognormal?
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad What Are Common Misconceptions About the Dirac Delta Function?
  • · Replies 25 ·
Replies
25
Views
4K
Graduate What is the use of Dirac delta function in quantum mechanics?
  • · Replies 8 ·
Replies
8
Views
4K