- #1

- 21

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- Homework Statement:
- What is the integral of the following equation?

- Relevant Equations:
- See below for the equation.

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- Thread starter gabriellelee
- Start date

- #1

- 21

- 1

- Homework Statement:
- What is the integral of the following equation?

- Relevant Equations:
- See below for the equation.

- #2

- 165

- 38

What are your limits of integral?

- #3

- 21

- 1

-∞ to +∞What are your limits of integral?

- #4

- 165

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So what do you think?-∞ to +∞

- #5

- 21

- 1

1?So what do you think?

- #6

- 165

- 38

Yes1?

- #7

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I know intuitively that the integral is 1, can you explain it to me mathematically?Yes

- #8

- 17,104

- 8,918

By definition:I know intuitively that the integral is 1, can you explain it to me mathematically?

$$\int_{-\infty}^{+\infty}\delta(x) f(x) dx = f(0)$$

- #9

wrobel

Science Advisor

- 801

- 532

- #10

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- #11

wrobel

Science Advisor

- 801

- 532

There is no sense to speak about integrals here. It is nothing more than symbol in this context.

A space of test functions is a conditional question. In different problems this space is introduced differently this depends of problem we consider . Loosely speaking, the smaller space of test functions the bigger space of generalized functions.

For example, consider a subspace of ##\mathcal D'(\mathbb{R})## that consists of generalized functions with compact support. Such generalized functions are naturally defined on ##C^\infty(\mathbb{R})##.

A space of test functions is a conditional question. In different problems this space is introduced differently this depends of problem we consider . Loosely speaking, the smaller space of test functions the bigger space of generalized functions.

For example, consider a subspace of ##\mathcal D'(\mathbb{R})## that consists of generalized functions with compact support. Such generalized functions are naturally defined on ##C^\infty(\mathbb{R})##.

Last edited:

- #12

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- #13

wrobel

Science Advisor

- 801

- 532

Just put

$$(\delta, \varphi):=\varphi(0),\quad \varphi\in C^\infty(\mathbb{R})$$

This is a continuous linear functional with respect the standard topology in ##C^\infty(\mathbb{R})##

- #14

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