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1 Discussion on Multi-Valued Functions Week 3 notes, Math 7651 1 Discussion on multi-valued functions Log function : Note that if z is written in its polar representation: z = r eiθ, where r = z and θ = arg z, then | | log z log r + i θ + 2inπ (1) ≡ for n Z is consistent with exp [log z] = z. This is easily shown through substitution. However,∈ (1) is not unambiguous on the plane. To make (1) a single-valued function (for a particular n) on the complex plane, one needs to restrict θ (θ , 2 π + θ ] , (2) ∈ 0 0 or [θ0 , 2 π + θ0) where θ0 is some constant (3) Remark 1 One might wonder if (1) provides the most general representation of the inverse of exp function. The following lemma answers this in the positive. Lemma 1 If log1 and log2 are two inverse functions of exp, then for any particular z, log z log z = 2 π i n (4) 1 − 2 for n Z. ∈ Proof. Let w = log z and w = log z. Then, exp(w w )= ew1 /ew2 = 1. Since for 1 1 2 2 1 − 2 ζ = x + iy, eζ 1 = 0 implies ex cos y 1=0and ex sin y = 0, we deduce y = 2nπ, and x = 0, implying−w w = ζ =2 π i n − 1 − 2 Remark 2 : It is possible that log z = log z for some set of z C, not for others. 1 2 ∈ For instance, if we define log z = log r + i θ, with θ in ( π, π] , (5) 1 − log2 z = log r + i θ, with θ in [0, 2 π) , (6) then clearly log1 z = log2 z for z in the first and second quadrant but not equal in the third and fourth quadrant. In general, n Z appearing in (4) varies with choice of z unless ∈ the choice of branch cut, i.e. choice of θ0, is the same for log1 and log2 functions. 1 Definition 2 The particular choice n =0 in (1), with θ = π in (2) defines a particular 0 − logarithmic function, called the principal branch of the logarithm, with the notation logp z or Log z. Definition 3 The set of all z for which log is undefined are called it’s branch points: (0 and in this case). ∞ Definition 4 The set of points across which log function undergoes a discontinuity for a particular restriction of the function in a plane is called a branch cut, i.e. z arg z = θ0 is the branch cut. { | } Definition 5 For specificied branch cut, i.e. θ0, the value of n is referred to as the branch of the logarithm. Remark 3 Branch cuts (and therefore discontinuities) of log function cannot be avoided if log z is to be uniquely defined in the complex plane. However, a branch cut is quite arbitrary. It can be completely avoided if we equate the domain through the polar pair (r, θ), with no restriction on θ. In that case, one can define uniquely log z = ln r + iθ , with θ ( , ) (7) ∈ −∞ ∞ In this representation, specification of z is not enough; we need to specify θ = θ0 at some z = z0 and obtain the value of θ at other values from continuity along a path connecting z0 to z. The identification of the complex domain, based on value of θ, is referred to a Riemann surface. Note that while log z is not continuous (across cuts), when restricted on the plane; it is indeed continuous on the Riemann surface. C Remark 4 Note that, for a general z1, z2 , logp (z1z2) = logp z1 + logp z2, though this is true on the Riemann surface, since∈ on a Riemann surfac6 e, with appropriate choice of the argument, i(θ1+θ2) log (z1 z2) = log r1r2 e = log r1 + log r2 + i θ1 + iθ2 = log z1 + log z2 Definition 6 A finite point z0 is a branch point of a function f(z) if for all sufficiently small ǫ > 0, f(z + ǫ ei(φ + 2π)) = f(z + ǫ eiφ) 0 6 0 iφ when f(z0 + ǫe ) changes continuously with φ. This means that if we circle around z = z0 in a sufficiently small circle, we do not to the same value of f. 2 Eg. log (z +1) at z = 1. If z +1 = ǫ eiφ, then log (z + 1) = log ǫ + iφ + 2 i n π. When φ is replaced by φ− + 2 π, we do not return to the same value. Definition 7 z = is a branch point of f(z) if f(1/w) has a branch point at w =0. ∞ Eg. log z has a branch point at since log 1/w = log w + 2inπ has a branch point at w = 0. However, log z+1 has∞ no branch points at−z = . z−1 ∞ Recall, we mentioned for a specific branch and branch-cut choice, it is not always true log(z1 z2) = log(z1) + log(z2) (8) However, if we do not assign any explicit restriction on arg (z1 z2), but instead define it as: arg (z1 z2) = arg z1 + arg z2 (9) then (8) will hold. For instance, if we take arg ( 1) = π, and apply definition (9) − arg ( 1)2 = 2 arg( 1) = 2 π and hence log ( 1)2 = 2 i π and not 0. In− defining a composite− function involving logarithm,− say. log (z2 1), it is convenient to place a 2 π interval restriction on arg (z 1) and arg (z +1), but−not on arg (z2 1); − − instead we require arg (z2 1) = arg (z 1) + arg(z + 1) , (10) − − in accordance to (9) Thus, if z 1= r eiθ1 and z = r eiθ2 , then, using (10), − 1 2 2 2 2 log (z 1) = log (r1 r2) + i arg(z 1) + 2inπ = log r1 + log r2 + i(θ1 + θ2)+2inπ − − (11) 2 If θ1θ2 ( π, π], the branch cuts for log (z 1) will be those shown in Fig.1. z = 1 and z = ∈ are− branch points as may be readily− checked. There is of course a denumerably± infinite∞ set of branches of the function, corresponding to differing integral values of the integer n. Alternately, if we restricted θ [0, 2 π), θ ( π, π], then the corresponding branch 1 ∈ 2 ∈ − cuts are as given below in Fig.1. Once again we have infinitely many branches characterized by integer n in (11) Consider another example of of a composite function involving logarithm: z 1 log − (12) z +1 3 Im z z θ θ 2 1 -1 1 Re z Figure 1: Branch cuts for log (z2 1) − i θ1 iθ2 In this case, let z 1 = r1 e and z +1 = r2 e . Then, if we put 2 π interval restriction on arg(z 1), but− avoid putting direct restrictions on arg z−1 itself, then if is possible ± z+1 to define z 1 arg − = arg (z 1) arg (z +1) (13) z +1 − − Then z 1 log − = log r log r + i(θ θ )+ 2 i n π (14) z +1 1 − 2 1 − 2 If θ1, θ2 ( π, π], then the corresponding branch cut is shown in Fig. 1. Note∈ that− there is no cut on the real axis, left of z = 1. The reason in this case − is that the two cuts, for log (z 1) and log (z + 1) have cancelled each other out. To see this, note that if we approach− from the top a point on the real axis to the left of -1, θ , θ = π; approaching from below, one gets θ , θ = π. In either case, (θ θ ) 1 2 1 2 − 1 − 2 appearing in (14) is equal to 0. So, for any fixed branch n, the function as defined in (14) has no discontinuity on the real axis to the left of 1 and hence no cut over there. Note if − we instead restrict θ1, θ2 [0, 2π), then the branch cut situation is still the same as in Fig. 1. In this case, the branch∈ cut cancellation is on the positive real axis to the right of the branch point z = 1. On the other hand if we restrict θ1 [0, 2π) and θ2 ( π, π], then the corresponding branch cut situation is described by Fig.∈ 1. ∈ − 4 Im z -1 1 Figure 2: Another choice of cuts for log (z2 1) − Im z -1 1 Re z z−1 Figure 3: Branch cuts for log z+1 It is to be noted that the branch cut situation described in Fig. 1 and Fig. 1 are not the only ones possible. One can have other 2 π interval restrictions on θ1 and θ2 that will give rise to other kinds of branch cuts for the function in (5). In practice, specific choices are made according to the requirement that the function be continuous in some part of the complex plane. zk for nonintegral k Definition 8 For nonintegral k zk = exp[k log z] = exp[k (log r + iθ + 2 i n π)] (15) 5 Im z -1 1 z−1 Figure 4: Another possible cut for log z+1 where for the purposes of uniqueness on the plane for fixed n, θ [θ , 2 π + θ ) or (θ , 2 π + θ ] (16) ∈ 0 0 0 0 Remark 5 Since k is not an integer, exp[2 inkπ] = 1; hence zk can have more than 6 one values for the same z, depending on n. A fixed choice of n in (15) and θ0 in (16) specifies as unique function zk on the plane.
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