- #1
geoduck
- 258
- 2
Suppose you have the integral:
[tex]\int^b_{-a}\frac{dx}{x+i\epsilon} [/tex]
where epsilon is a tiny positive number. The integral should be a log shouldn't it?
[tex]=log[b+i\epsilon]-log[-a+i\epsilon]=log-(log[a]+...)[/tex]
where the ... is 'i' times the angle of the point -a+iε. Doesn't this depend on where you put your branch cut? If your cut goes in the direction of the positive imaginary axis, then the angle is -pi (the angle goes from 0 in the direction of the positive real axis, to pi/2, then it drops discontinuously to -3pi/2 as it crosses the cut). However, if you put your cut on the negative real axis, then the angle is +pi. If you put your cut on the positive real axis, the angle is also +pi.
How do you know which cut to use?
[tex]\int^b_{-a}\frac{dx}{x+i\epsilon} [/tex]
where epsilon is a tiny positive number. The integral should be a log shouldn't it?
[tex]=log[b+i\epsilon]-log[-a+i\epsilon]=log-(log[a]+...)[/tex]
where the ... is 'i' times the angle of the point -a+iε. Doesn't this depend on where you put your branch cut? If your cut goes in the direction of the positive imaginary axis, then the angle is -pi (the angle goes from 0 in the direction of the positive real axis, to pi/2, then it drops discontinuously to -3pi/2 as it crosses the cut). However, if you put your cut on the negative real axis, then the angle is +pi. If you put your cut on the positive real axis, the angle is also +pi.
How do you know which cut to use?