Mathematical Methods for Physics and Engineering - Matematica.NET

24.5 MULTIVALUED FUNCTIONS AND BRANCH CUTSOn the RHS let us write t as follows:t = r exp[i(θ +2kπ)],where k is an integer. We then obtain[ ]1t 1/n (θ +2kπ)=exp ln r + in n[ ]= r 1/n (θ +2kπ)exp i ,nwhere k =0, 1,...,n− 1; for other values of k we simply recover the roots already found.Thus t has n distinct nth roots. ◭24.5 Multivalued functions and branch cutsIn the definition of an analytic function, one of the conditions imposed wasthat the function is single-valued. However, as shown in the previous section, thelogarithmic function, a complex power and a complex root are all multivalued.Nevertheless, it happens that the properties of analytic functions can still beapplied to these and other multivalued functions of a complex variable providedthat suitable care is taken. This care amounts to identifying the branch points ofthe multivalued function f(z) in question. If z is varied in such a way that itspath in the Argand diagram forms a closed curve that encloses a branch point,then, in general, f(z) will not return to its original value.For definiteness let us consider the multivalued function f(z) =z 1/2 and expressz as z = r exp iθ. From figure 24.1(a), it is clear that, as the point z traverses anyclosed contour C that does not enclose the origin, θ will return to its originalvalue after one complete circuit. However, for any closed contour C ′ that doesenclose the origin, after one circuit θ → θ +2π (see figure 24.1(b)). Thus, for thefunction f(z) =z 1/2 , after one circuitr 1/2 exp(iθ/2) → r 1/2 exp[i(θ +2π)/2] = −r 1/2 exp(iθ/2).In other words, the value of f(z) changes around any closed loop enclosing theorigin; in this case f(z) →−f(z). Thus z = 0 is a branch point of the functionf(z) =z 1/2 .We note in this case that if any closed contour enclosing the origin is traversedtwice then f(z) =z 1/2 returns to its original value. The number of loops arounda branch point required for any given function f(z) to return to its original valuedepends on the function in question, and for some functions (e.g. Ln z, which alsohas a branch point at the origin) the original value is never recovered.In order that f(z) may be treated as single-valued, we may define a branch cutin the Argand diagram. A branch cut is a line (or curve) in the complex planeand may be regarded as an artificial barrier that we must not cross. Branch cutsare positioned in such a way that we are prevented from making a complete835