Green function as distributions

[matematikuvol]

If we have Green function

$$g(x,s)=exp[-\int^x_s p(z)dz]$$ we want to think about that as distribution so we multiply it with Heaviside step function

$$g(x,s)=H(x-s)exp[-\int^x_s p(z)dz]$$

Why we can just multiply it with step function and tell that the function is the same. Tnx for the answer.

 

[matematikuvol]

To be more precise.

If I say solution to eq is

$$u(x)=\int^{x}_0g(x,s)f(s)ds$$

where $$g=e^{-\int^x_yp(z)dz}$$

Then if I define

$$g=H(x-s)e^{-\int^x_sp(z)dz}$$

is then

$$u(x)=\int^{\infty}_0g(x,s)f(s)ds$$

and what is Green function this $$g=e^{-\int^x_yp(z)dz}$$ or this $$g=H(x-s)e^{-\int^x_sp(z)dz}$$?

 

[matematikuvol]

Can you help me?