Delta function defined for complez values

In summary, the conversation discusses the possibility of defining the Dirac delta function for complex values, specifically using test functions and scaling properties. It is noted that there should not be a problem using the delta function with a complex constant or multiplying x by i, but caution should be taken when applying it to a problem. The delta function is only meaningful when used under integral signs.
  • #1
mhill
189
1
is there a form to define the dirac delta function for complex values ? i mean

[tex] \delta (x-a-bi) [/tex] or [tex] \delta (-ix) [/tex]

using 'test functgions' i get that they converge nowhere (always infinite) which makes no sense at all, using scalling properties we could define

[tex] \delta (ix) = \delta(x) [/tex] since modulus of 'i' is just one but i am not completely sure.
 
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  • #2
mhill said:
is there a form to define the dirac delta function for complex values ? i mean

[tex] \delta (x-a-bi) [/tex] or [tex] \delta (-ix) [/tex]

using 'test functgions' i get that they converge nowhere (always infinite) which makes no sense at all, using scalling properties we could define

[tex] \delta (ix) = \delta(x) [/tex] since modulus of 'i' is just one but i am not completely sure.

In using the delta function, the domain is the real line. There is no problem with using it with a complex constant or even multiplying x by i. Just be careful about what problem you are trying to solve. Remeber the delta function makes sense only under integral signs.
 

Related to Delta function defined for complez values

1. What is the definition of the Delta function for complex values?

The Delta function for complex values is a mathematical function that is defined as a generalized function on the complex plane. It is often denoted as δ(z), where z is a complex variable. The function has the property that it is zero for all values of z except when z=0, where it is infinite. The integral of the Delta function over any region containing z=0 is equal to 1.

2. How is the Delta function for complex values related to the real Delta function?

The Delta function for complex values is an extension of the real Delta function, which is also known as the Dirac delta function. The real Delta function is defined on the real line and has similar properties as the Delta function for complex values. However, the complex Delta function takes into account the complex plane and has a slightly different definition and properties.

3. How is the Delta function for complex values used in mathematics and physics?

The Delta function for complex values has various applications in mathematics and physics. It is often used in complex analysis, where it helps in solving problems involving complex variables and integrals. In physics, it is used to model point sources of energy or mass in fields such as electromagnetism and quantum mechanics.

4. Can the Delta function for complex values be generalized to higher dimensions?

Yes, the Delta function for complex values can be extended to higher dimensions, just like the real Delta function. In higher dimensions, it becomes a distribution and is often denoted as δ(z1, z2, ..., zn), where z1, z2, ..., zn are complex variables. It still has the property that it is zero everywhere except at the origin, where it is infinite, and its integral over any region containing the origin is equal to 1.

5. What is the connection between the Delta function for complex values and Fourier transforms?

The Delta function for complex values plays a crucial role in the theory of Fourier transforms. It is used to define the inverse Fourier transform of a function, which involves integrating the function multiplied by the Delta function over the complex plane. This allows us to express a function in terms of its Fourier transform and vice versa, making it a powerful tool in solving differential equations and other problems in mathematics and physics.

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