- #1
binbagsss
- 1,265
- 11
Homework Statement
I have
##\int dx \int dy \delta (x^{2}+y^{2}-E) ## [1]
I have only seen expressions integrating over ##\delta## where the ##x## or the ##y## appear seperately as well as in the delta function and so you can just replace e.g ##y^2 = - x^{2} +E## then integrate over ##\int dy ##, ##\int dx ##.
Homework Equations
see above
The Attempt at a Solution
I am unsure how to do this, here is my working so far:
##x^{2}=E-y^{2} ##
=>
## 2x dx = E - 2y dy ##
## 2(E-y^{2})^{1/2} dx = E-2y dy ##
## dx = \frac{E}{ 2(E-y^{2})^{1/2}} - \frac{y}{ (E-y^{2})^{1/2}} dy ##
## dx = \frac{E}{ 2(E-y^{2})^{1/2}} - (E-y^{2})^{1/2} ##
So can this step in affect replace the integrating over ##x## part of the delta function, to replace ##dx## so that
[1] reduces to ## \int dy \frac{E}{ 2(E-y^{2})^{1/2})} - (E-y^{2})^{1/2} ##
Am i on the right track?
Many thanks in advance.