We begin by solving the problem from the last instalment:
Problem: Show that Log z is well defined and single-valued when z not equal 0
(Hint: use z = r exp(i0)
Solution: z = r exp(i0) = r exp(0) = r(1) = r
Then: Log z = ln r
so Log r = ln r
Now, a branch of a multiple-valued function f is any single-valued function F that is analytic at some domain at a point z for which the value F(z) = f(z). (Being “analytic” means only that F cannot take on random values of f), Then for each fixed real number, k, the single-valued function
e.g. log z= ln r + i(q) (r > 0, k < q < k + 2 p)
is a branch of the multiple –valued function:
log z= ln r + i(q)
Then the function:
Log z = ln r + iQ (r > 0, - p < Q < p)
Is called the principal branch.
Now, a branch cut is a portion of a line or curve that is introduced in order to define a branch F of a multiple –valued function f. Points on the branch cut for F are singular points of F, and any point that is common to all branch cuts of f is called a branch point.
Example: For a sunspot model
In the diagram shown, a multipole sunspot model is depicted such that the most intense magnetic field locus is centered in the largest sunspot and is "line-tied" , i.e. it represents the anchored ‘foot’ of one magnetic arch with the other foot connected to a 2nd polarity (minor spot) executing proper motion around it.
In this case the dipole undergoes an approximate 28 degree proper motion over time interval (t2 - t1) during which the (-) polarity footpoint is displaced from a1 to a2. This translates the location of the field line f1 from roughly a1 = (-3k – 3ki) where each k = 10 4 km, to a2 (-k – 4ki). The neutral line or magnetic inversion line itself passes through the sunspot along a defined Re x -axis. Here, we designate the ray Q = 2 p, as a branch cut along the positive Re-axis. This is really the branch cut for the functions f and F to be worked out. The origin (x=0, y=0) is putatively the branch point for the system.
In effect, the branch cut is a barrier separating the domain of one branch of a function from a different branch. A multiple –valued function, meanwhile, is discontinuous at every point of a branch cut for the function. Setting the origin for our sunspot system evidently means setting branch points for branches of a multiple-valued logarithmic function too.
Problems for the Math Maven:
1) Using the diagram for the multipole sunspot shown, identify the principal branch for the given branch cut. (Hint: express as a function of r, Q)
2) Consider the complex z-plane, i.e. z(x, iy) featuring concentric circles of differing radii a = 0.5, 1, and 2 such that the equation: êz ê = r = a applies. Show these concentric circles and then show at least three lines on the same coordinate system which satisfy q = a . (where a is any designated angle)
3) For the multipole sunspot diagram shown, assume the ray Q1(a1,t1) = 6 p/5 and Q2(a2,t2) = 7 p/5 for a point z defined in proper motion about the origin, anchored in the major spot. Can you write a function describing this proper motion? (Hint: w = u + iv = ln z = ln r + iq, , so that u = ln r and v = q, .). Be sure to cite any assumptions made.