| 52 | | Now $B_\perp$ is unchanged under rotations about the jet axis, however $B_x$ will be given by $B_\phi \cos \phi$ |
| | 52 | Now $B_\perp$ is unchanged under rotations about the jet axis, however $B_x$ will be given by $B_\phi \cos \phi$ or $B_x = B_\phi \frac{x_i}{\sqrt{x_i^2+(s \cos(\theta))^2}}$ |
| | 53 | |
| | 54 | And finally we can change the path integral to an x integral under the substitution |
| | 55 | |
| | 56 | $x=\sqrt{x_i^2+s^2}$ |
| | 57 | |
| | 58 | or |
| | 59 | |
| | 60 | $s = \sqrt{x^2-x_i^2}$ |
| | 61 | |
| | 62 | and |
| | 63 | |
| | 64 | $ds/dx = \frac{x}{\sqrt{x^2-x_i^2}}$ |
| | 65 | |
| | 66 | This gives |
| | 67 | |
| | 68 | |
| | 69 | $\int_{x} { f(B'(x, y_i,0)) \frac{x^2}{x^2-x_i^2} dx}$ |
| | 70 | |
| | 71 | and $B_x'(x,y_i,0)=B_\phi(x,y_i,0) \frac{x_i}{x}$ |
| | 72 | |
| | 73 | |
