analytic continuation
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Word-processor note dated 1.11.15 (initials PhL) working through analytic continuation of a function with a branch point at z0. It tracks the phases of z and (z-z0) when moving from real z>0 to z<0 just above a cut, giving factors like e^{iπ/2}=i. It then applies this to the quadratic case 4ac-b^2>0 with c>0, continuing to c<0. Equations and figures are partly lost in extraction.
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This is the Title PhL 1.11.15
Analytic Continuation
Consider this generic problem.
Let
f(z) = with z0 > 0
For complex z write
z = |z|eiθ
(z-z0) = |z-z0| eiφ
We start with real z = a where a > z0 so (a-z0) > 0. In this case
a = |a|ei0 so θ = 0
(a-z0) = |a-z0|ei0 so φ = 0
Now let z = -a which we define to be -a+iε just above the cut shown below. Then
-a = |a|eiπ = a eiπ
(-a-z0) = |-a-z0| eiπ = |a+z0|eiπ = (a+z0)eiπ
Then
= eiπ/2 = i
= eiπ/2 = i
Start with Case 2 of (2.5) where 4ac - b2 > 0 and c > 0. Assume further (for the moment) that a > 0. Then define z ≡ 4ac. We shall treat z as a complex variable and analytically continue from z > 0 to z<0. The function has a branch point at z = b2 whose cut we take off to the left. We shall associate the quantity z > 0 with (4ac)ei0 and z < 0 with (4ac)eiπ , a point above the cut. We then smoothly continue from z > 0 to z < 0 as shown in this drawing,
For z > 0 (c > 0) we have θ = 0 and α = 0.
For z < 0 (c < 0) we have θ = π and α = π.
c = |c| eiπ = eiπ/2
(4ac-b2) = |4ac-b2| eiπ