The complex inverse trigonometric and hyperbolic functions Howard E. Haber

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1 The complex inverse trigonometric and hyperbolic functions Howard E. Haber Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA 95064, USA September 10, 01 Abstract In these notes, we examine the inverse trigonometric and hyperbolic functions, where the arguments of these functions can be complex numbers. The multi-valued functions are defined in terms of the complex logarithm. We also carefully define the corresponding single-valued principal values of the inverse trigonometric and hyperbolic functions following the conventions employed by the computer algebra software system, Mathematica 8. 1 Introduction The inverse trigonometric and hyperbolic functions evaluated in the complex plane are multi-valued functions (see e.g. ref. 1). In many applications, it is convenient to define the corresponding single-valued functions, called the principal values of the inverse trigonometric and hyperbolic functions, according to some convention. Different conventions appear in various reference books. In these notes, we shall follow the conventions employed by the computer algebra software system, Mathematica 8, which are outlined in section..5 of ref.. The principal value of a multi-valued complex function f(z) of the complex variable z, which we denote by F(z), is continuous in all regions of the complex plane, except on a specific line (or lines) called branch cuts. The function F(z) has a discontinuity when z crosses a branch cut. Branch cuts end at a branch point, which is unambiguous for each function F(z). But the choice of branch cuts is a matter of convention. Thus, if mathematics software is employed to evaluate the function F(z), you need to know the conventions of the software for the location of the branch cuts. The mathematical software needs to precisely define the principal value of f(z) in order that it can produce a unique answer when the user types in F(z) for a particular complex number z. There are often different possible candidates for F(z) that differ only in the values assigned to them when z lies on the branch cut(s). These notes provide a careful discussion of these issues as they apply to the complex inverse trigonometric and hyperbolic functions. The simplest example of a multivalued function is the argument of a complex number z, denoted by arg z. In these notes, the principal value of the argument of the complex number z, denoted by Argz, is defined to lie in the interval π < Argz π. That is, Argz is single-valued and is continuous at all points in the complex plane 1

2 excluding a branch cut along the negative real axis. In Appendix A, a detailed review of the properties of arg z and Argz are provided. The properties of Argz determine the location of the branch cuts of the principal values of the logarithm the square root functions. The complex logarithm and generalized power functions are reviewed in Appendix B. If f(z) is expressible in terms of the logarithm the square root functions, then the definition of the principal value of F(z) is not unique. However given a specific definition of F(z) in terms of the principal values of the logarithm the square root functions, the locations of the branch cuts of F(z) are inherited from that of Argz and are thus uniquely determined. The inverse trigonometric functions: arctan and arccot We begin by examining the solution to the equation z = tanw = sinw cosw = 1 ( ) e iw e iw = 1 i e iw +e iw i ( ) e iw 1. e iw +1 We now solve for e iw, iz = eiw 1 e iw +1 = e iw = 1+iz 1 iz. Taking the complex logarithm of both sides of the equation, we can solve for w, w = 1 ( ) 1+iz i ln. 1 iz The solution to z = tanw is w = arctanz. Hence, arctanz = 1 ( ) 1+iz i ln 1 iz (1) Since the complex logarithm is a multi-valued function, it follows that the arctangent function is also a multi-valued function. Using the definition of the multi-valued complex logarithm, arctanz = 1 i Ln 1+iz 1 iz +1 [ Arg ( ) ] 1+iz +πn, n = 0, ±1, ±,..., () 1 iz where Arg is the principal value of the argument function. Similarly, z = cotw = cosw ( ) ( ) i(e iw sinw = +e iw i(e iw +1 =. e iw e iw e iw 1

3 Again, we solve for e iw, iz = eiw +1 e iw 1 = e iw = iz 1 iz +1. Taking the complex logarithm of both sides of the equation, we conclude that w = 1 ( ) iz 1 i ln = 1 ( ) z +i iz +1 i ln, z i after multiplying numerator and denominator by i to get a slightly more convenient form. The solution to z = cotw is w = arccotz. Hence, arccotz = 1 ( ) z +i i ln (3) z i Thus, the arccotangent function is a multivalued function, arccotz = 1 [ ( ) ] i Ln z +i z +i z i + 1 Arg +πn, n = 0, ±1, ±,..., (4) z i Using the definitions given by eqs. (1) and (3), the following relation is easily derived: ( ) 1 arccot(z) = arctan. (5) z Note that eq. (5) can be used as the definition of the arccotangent function. It is instructive to derive another relation between the arctangent and arccotangent functions. First, we first recall the property of the multi-valued complex logarithm, ln(z 1 z ) = ln(z 1 )+ln(z ), (6) as a set equality [cf. eq. (B.14)]. It is convenient to define a new variable, It follows that: arctanz +arccotz = 1 i v = i z i+z, = 1 v = z +i z i. (7) [ ( lnv +ln 1 )] = 1 ( ) v v i ln = 1 v i ln( 1). Since ln( 1) = i(π +πn) for n = 0,±1,±..., we conclude that arctanz +arccotz = 1 π +πn, for n = 0,±1,±,... (8) Finally, we mention two equivalent forms for the multi-valued complex arctangent and arccotangent functions. Recall that the complex logarithm satisfies ( ) z1 ln = lnz 1 lnz, (9) z 3

4 where this equation is to be viewed as a set equality [cf. eq. (B.15). Thus, the multivalued arctangent and arccotangent functions given in eqs. (1) and (5), respectively, are equivalent to arctanz = 1 [ ] ln(1+iz) ln(1 iz), (10) i arccotz = 1 [ ( ln 1+ i ) ( ln 1 i )], (11) i z z 3 The principal values Arctan and Arccot It is convenient to define principal values of the inverse trigonometric functions, which are single-valued functions, which will necessarily exhibit a discontinuity across branch cuts in the complex plane. In Mathematica 8, the principal values of the complex arctangent and arccotangent functions, denoted by Arctan and Arccot respectively (with a capital A), are defined by employing the principal values of the complex logarithms in eqs. (10) and (11), Arctanz = 1 i [ ] Ln(1+iz) Ln(1 iz), z ±i (1) and Arccotz = Arctan ( ) 1 = 1 [ ( Ln 1+ i ) ( Ln 1 i )],z ±i, z 0 (13) z i z z Notethatthepointsz = ±iareexcluded fromtheabovedefinitions, asthearctangent and arccotangent are divergent at these two points. The definition of the principal value of the arccotangent given in eq. (13) is deficient in one respect since it is not well-defined at z = 0. We shall address this problem shortly. One useful feature of the definitions eqs. (1) and (13) is that they satisfy: Arctan( z) = Arctanz, Arccot( z) = Arccotz. Because the principal value of the complex logarithm Ln does not satisfy eq. (9) in all regions of the complex plane, it follows that the definitions of the complex arctangent and arccotangent functions adopted by Mathematica 8 do not coincide with alternative definitions that employ the principal value of the complex logarithms in eqs. (1) and (4) [for further details, see Appendix C]. First, we shall identify the location of the branch cuts by identifying the lines of discontinuity of the principal values of the complex arctangent and arccotangent 4

5 functions in the complex plane. The principal value of the complex arctangent function is single-valued for all complex values of z, excluding the two branch points at z ±i. Moreover, the the principal-valued logarithms, Ln(1 ± iz) are discontinuous as z crosses the lines 1 ± iz < 0, respectively. We conclude that Arctanz must be discontinuous when z = x+iy crosses lines on the imaginary axis such that x = 0 and < y < 1 and 1 < y <. (14) These two lines thatliealong theimaginaryaxis arethebranch cuts ofarctanz. Note that Arctan z is single-valued on the branch cut itself, since it inherits this property from the principal value of the complex logarithm. Likewise, the principal value of the complex arccotangent function is single-valued for all complex z, excluding the branch points z ±i. Moreover, the the principalvalued logarithms, Ln ( 1± z) i are discontinuous as z crosses the lines 1 ± i < 0, z respectively. We conclude that Arccot z must be discontinuous when z = x + iy crosses the branch cut located on the imaginary axis such that In particular, due to the presence of the branch cut, lim x 0 x = 0 and 1 < y < 1. (15) Arccot(x+iy) lim +Arccot(x+iy), for 1 < y < 1, x 0 for real values of x, where 0 + indicates that the limit is approached from positive real axis and 0 indicates that the limit is approached from negative real axis. If z 0, eq. (13) provides unique values for Arccot z for all z ±i in the complex plane, including points on the branch cut. Using eq. (1), one can easily show that if z is a non-zero complex number infinitesimally close to 0, then Arccot z = z 0,z π, for Re z > 0, π, for Re z = 0 and Im z < 0, π, for Re z < 0, π, for Re z = 0 and Im z > 0. It is now apparent why z = 0 is problematical in eq. (13), since lim z 0 Arccotz is not uniquely defined by eq. (16). If we wish to define a single-valued arccotangent function, then we must separately specify the value of Arccot(0). Mathematica 8 supplements the definition of the principal value of the complex arccotangent given in eq. (13) by declaring that Arccot(0) = 1 π. (17) With the definitions given in eqs. (1), (13) and (17), Arctan z and Arccot z are single-valued functions in the entire complex plane, excluding the branch points z = ±i, and are continuous functions as long as the complex number z does not cross the branch cuts specified in eqs. (14) and (15), respectively. 5 (16)

6 Having defined precisely the principal values of the complex arctangent and arccotangent functions, let us check that they reduce to the conventional definitions when z is real. First consider the principal value of the real arctangent function, which satisfies, 1 1π Arctanx 1 π, for x, (18) where x is a real variable. The definition given by eq. (1) does reduce to the conventional definition of the principal value of the real-valued arctangent function when z is real. In particular, for real values of x, Arctanx = 1 i [ ] Ln(1+ix) Ln(1 ix) = 1 [ ] Arg(1+ix) Arg(1 ix), (19) after noting that Ln 1+ix = Ln 1 ix = 1 Ln(1+x ). Geometrically, the quantity Arg(1+ix) Arg(1 ix) is the angle between the complex numbers 1+ix and 1 ix viewed as vectors lying in the complex plane. This angle varies between π and π over the range < x <. Moreover, the values ±π are achieved in the limit as x ±, respectively. Hence, we conclude that the principal interval of the realvalued arctangent function is indeed given by eq. (18). For all possible values of x excluding x =, one can check that it is permissible to subtract the two principalvalued logarithms (or equivalently the two Arg functions) using eq. (9). In the case of x, eq. (B.1) yields Arg(1 + ix) Arg(1 ix) π, corresponding to N = 1 inthenotationofeq. (A.1). Hence, anextra termappearswhen combining the two logarithms that is equal to πin = πi. The end result is, Arctan( ) = 1 i [ln( 1) πi] = 1 π, as required. As a final check, we can use the results of Tables 1 and given in Appendix A to conclude that Arg(a+bi) = Arctan(b/a) for a > 0. Setting a = 1 and b = x then yields: Arg(1+ix) = Arctanx, Arg(1 ix) = Arctan( x) = Arctanx. Subtracting these two results yields eq. (19). In contrast to the real arctangent function, there is no generally agreed definition for the principal range of the real-valued arccotangent function. However, a growing consensus among computer scientists has led to the following choice for the principal range of the real-valued arccotangent function, 1π < Arccotx 1 π, for x, (0) 1 In defining the principal value of the real arctangent, we follow the conventions of Keith B. Oldham, Jan Myland and Jerome Spanier, An Atlas of Functions (Springer Science, New York, 009), Chapter 35. 6

7 where x is a real variable. Note that the principal value of the arccotangent function does not include the endpoint 1 π [contrast this with eq. (18) for Arctan]. The reason for this behavior is that Arccotx is discontinuous at x = 0, with lim x 0 Arccotx = 1π, lim x 0 +Arccotx = 1 π, (1) as a consequence of eq. (16). In particular, eq. (0) corresponds to the convention in which Arccot(0) = 1 π [cf. eq. (17)]. Thus, as x increases from negative to positive values, Arccotx never reaches 1π but jumps discontinuously to 1 π at x = 0. Finally, we examine the the analog of eq. (8) for the corresponding principal values. Employing the Mathematica 8 definitions for the principal values of the complex arctangent and arccotangent functions, we find that 1π, for Re z > 0, 1π, for Re z = 0, and Im z > 1 or 1 < Im z 0, Arctanz +Arccotz = 1 π, for Re z < 0, 1 π, for Re z = 0, and Im z < 1 or 0 < Im z < 1. () The derivation of this result will be given in Appendix D. In Mathematica, one can confirm eq. () with many examples. The relations between the single-valued and multi-valued functions can be summarized by: arctanz = Arctanz +nπ, n = 0, ±1, ±,, arccotz = Arccotz +nπ, n = 0, ±1, ±,. Note that we can use these relations along with eq. () to confirm the result obtained in eq. (8). 4 The inverse trigonometric functions: arcsin and arccos The arcsine function is the solution to the equation: z = sinw = eiw e iw. i Letting v e iw and multiplying the resulting equation by v yields the quadratic equation, v izv 1 = 0. (3) The solution to eq. (3) is: v = iz +(1 z ) 1/. (4) 7

8 Since z is a complex variable, (1 z ) 1/ is the complex square-root function. This is a multi-valued function with two possible values that differ by an overall minus sign. Hence, we do not explicitly write out the ± sign in eq. (4). To avoid ambiguity, we shall write v = iz +(1 z ) 1/ = iz +e 1 ln(1 z ) = iz +e 1 [Ln 1 z +iarg(1 z )] In particular, note that = iz + 1 z 1/ e i arg(1 z). e i arg(1 z) = e i Arg(1 z) e inπ = ±e i Arg(1 z), for n = 0,1, which exhibits the two possible sign choices. By definition, v e iw, from which it follows that w = 1 i lnv = 1 ( ) i ln iz + 1 z 1/ e i arg(1 z ). The solution to z = sinw is w = arcsinz. Hence, arcsinz = 1 ( ) i ln iz + 1 z 1/ e i arg(1 z ) The arccosine function is the solution to the equation: Letting v e iw, we solve the equation z = cosw = eiw +e iw. v+ 1 v = z. Multiplying by v, one obtains a quadratic equation for v, The solution to eq. (5) is: v zv +1 = 0. (5) v = z +(z 1) 1/. Following the same steps as in the analysis of arcsine, we write w = arccosz = 1 i lnv = 1 i ln[ z +(z 1) 1/], (6) where (z 1) 1/ is the multi-valued square root function. More explicitly, arccosz = 1 ( ) i ln z + z 1 1/ e i arg(z 1). (7) 8

9 It is sometimes more convenient to rewrite eq. (7) in a slightly different form. Recall that arg(z 1 z ) = argz +argz, (8) as a set equality [cf. eq. (A.10). We now substitute z 1 = z and z = 1 into eq. (8) and note that arg( 1) = π +πn (for n = 0,±1,±,...) and argz = argz +πn as a set equality. It follows that as a set equality. Thus, and we can rewrite eq. (6) as follows: arg( z) = π +argz, e i arg(z 1) = e iπ/ e i arg(1 z ) = ie i arg(1 z ), which is equivalent to the more explicit form, arccosz = 1 i ln ( z +i 1 z ), (9) arccosz = 1 ( ) i ln z +i 1 z 1/ e i arg(1 z ) The arcsine and arccosine functions are related in a very simple way. Using eq. (4), i v = i iz + 1 z = i( iz + 1 z ) (iz + 1 z )( iz + 1 z ) = z +i 1 z, which we recognize as the argument of the logarithm in the definition of the arccosine [cf. eq. (9)]. Using eq. (6), it follows that arcsinz +arccosz = 1 [ ( )] i lnv +ln = 1 ( ) iv i v i ln = 1 v i lni. Since lni = i( 1 π +πn) for n = 0,±1,±..., we conclude that arcsinz +arccosz = 1 π +πn, for n = 0,±1,±,... (30) 5 The principal values Arcsin and Arccos In Mathematica 8, the principal value of the arcsine function is obtained by employing the principal value of the logarithm and the principle value of the square-root function (which corresponds to employing the principal value of the argument). Thus, Arcsinz = 1 ( ) i Ln iz + 1 z 1/ e i Arg(1 z ). (31) 9

10 It is convenient to introduce some notation for the the principle value of the squarerootfunction. Consider themultivalued squarerootfunction, denotedbyz 1/. Henceforth, we shall employ the symbol z to denote the single-valued function, z = z e 1 Arg z, (3) where z denotes the unique positive squared root of the real number z. In this notation, eq. (31) is rewritten as: Arcsinz = 1 i Ln ( iz + 1 z ) One noteworthy property of the principal value of the arcsine function is To prove this result, it is convenient to define: Then, v = iz + 1 z, Arcsinz = 1 i Lnv, (33) Arcsin( z) = Arcsin z. (34) 1 v = 1 iz + 1 z = iz + 1 z. (35) Arcsin( z) = 1 ( ) 1 i Ln. v The second logarithm above can be simplified by making use of eq. (B.3), { Ln(z)+πi, if z is real and negative, Ln(1/z) = Ln(z), otherwise. (36) In Appendix E, we prove that v can never be real and negative. Hence it follows from eq. (36) that Arcsin( z) = 1 ( ) 1 i Ln = 1 Lnv = Arcsinz, v i as asserted in eq. (34). We now examine the principal value of the arcsine for real-valued arguments such that 1 x 1. Setting z = x, where x is real and x 1, Arcsinx = 1 ( i Ln ix+ ) 1 x = 1 [ Ln ix+ ( +iarg 1 x i ix+ )] 1 x = Arg (ix+ ) 1 x, for x 1, (37) since ix + 1 x is a complex number with magnitude equal to 1 when x is real with x 1. Moreover, ix + 1 x lives either in the first or fourth quadrant of the complex plane, since Re(ix+ 1 x ) 0. It follows that: π Arcsinx π, for x 1. 10

11 In Mathematica 8, the principal value of the arccosine is defined by: Arccosz = 1 π Arcsinz. (38) We demonstrate below that this definition is equivalent to choosing the principal value of the complex logarithm and the principal value of the square root in eq. (9). That is, Arccosz = 1 i Ln ( z +i 1 z ) To verify that eq. (38) is a consequence of eq. (39), we employ the notation of eq. (35) to obtain: Arcsinz +Arccosz = 1 [ ( ) ( )] 1 i Ln v +Ln +iargv +iarg i v v ( ) i = Arg v +Arg. (40) v It is straightforward to check that: ( ) i Arg v +Arg v = 1 π, for Re v 0. However in Appendix F, we prove that Re v Re (iz+ 1 z ) 0 for all complex numbers z. Hence, eq. (40) yields: Arcsinz +Arccosz = 1 π, as claimed. We now examine the principal value of the arccosine for real-valued arguments such that 1 x 1. Setting z = x, where x is real and x 1, [ Ln x+i ( +iarg 1 x x+i )] 1 x Arccosx = 1 i Ln ( x+i 1 x ) = 1 i (39) = Arg (x+i ) 1 x, for x 1, (41) since x + i 1 x is a complex number with magnitude equal to 1 when x is real with x 1. Moreover, x+i 1 x lives either in the first or second quadrant of the complex plane, since Im(x+i 1 x ) 0. It follows that: 0 Arccosx π, for x 1. The principal value of the complex arcsine and arccosine functions are singlevalued for all complex z. The choice of branch cuts for Arcsinz and Arccosz must coincide in light of eq. (38). Moreover, due to the standard branch cut of the principal value square root function, it follows that Arcsinz is discontinuous when z = x+iy One can check that the branch cut of the Ln function in eq. (33) is never encountered for any finite value of z. For example, in the case of Arcsinz, the branch cut of Ln can only be reached if iz + 1 z is real and negative. But this never happens since if iz + 1 z is real then z = iy for some real value of y, in which case iz + 1 z = y + 1+y > 0. 11

12 crosses lines on the real axis such that 3 y = 0 and < x < 1 and 1 < x <. (4) These two lines comprise the branch cuts of Arcsinz and Arccosz; each branch cut ends at a branch point located at x = 1 and x = 1, respectively (although the square root function is not divergent at these points). 4 To obtain the relations between the single-valued and multi-valued functions, we first notice that the multi-valued nature of the logarithms imply that arcsin z can take on the values Arcsinz+πn and arccosz can take on the values Arccosz+πn, where n is any integer. However, we must also take into account the fact that (1 z ) 1/ can take on two values, ± 1 z. In particular, arcsinz = 1 i ln(iz ± 1 z ) = 1 ( ) i ln 1 iz = 1 [ ln( 1) ln(iz ] 1 z 1 z i ) = 1 i ln(iz 1 z )+(n+1)π, where n is any integer. Likewise, arccosz = 1 i ln(z ±i 1 z ) = 1 ( ) i ln 1 z i = 1 1 z i ln(z i 1 z )+πn, where n is any integer. Hence, it follows that arcsinz = ( 1) n Arcsinz +nπ, n = 0, ±1, ±,, (43) arccosz = ±Arccosz +nπ, n = 0, ±1, ±,, (44) where either ±Arccosz can be employed to obtain a possible value of arccosz. In particular, the choice of n = 0 in eq. (44) implies that: arccosz = arccosz, (45) which should be interpreted as a set equality. Note that one can use eqs. (43) and (44) along with eq. (38) to confirm the result obtained in eq. (30). 6 The inverse hyperbolic functions: arctanh and arccoth Consider the solution to the equation z = tanhw = sinhw coshw = ( ) e w e w = e w +e w ( ) e w 1. e w +1 3 Note that for real w, we have sinw 1 and cosw 1. Hence, for both the functions w = Arcsinz and w = Arccosz, it is desirable to choose the branch cuts to lie outside the interval on the real axis where Re z 1. 4 The functions Arcsinz and Arccosz also possess a branch point at the point of infinity (which is defined more precisely in footnote 5). This can be verified by demonstrating that Arcsin(1/z) and Arccos(1/z) possess a branch point at z = 0. For further details, see e.g. Section 58 of ref 3. 1

13 We now solve for e w, z = ew 1 e w +1 = e w = 1+z 1 z. Taking the complex logarithm of both sides of the equation, we can solve for w, w = 1 ( ) 1+z ln. 1 z The solution to z = tanhw is w = arctanhz. Hence, arctanhz = 1 ( ) 1+z ln 1 z (46) Similarly, by considering the solution to the equation z = cothw = coshw ( ) ( ) e w sinhw = +e w e w +1 =. e w e w e w 1 we end up with: The above results then yield: arccothz = 1 ( ) z +1 ln z 1 arccoth(z) = arctanh ( ) 1, z (47) as a set equality. Finally, we note the relation between the inverse trigonometric and the inverse hyperbolic functions: arctanhz = iarctan( iz), arccothz = iarccot(iz). As in the discussion at the end of Section 1, one can rewrite eqs. (46) and (47) in an equivalent form: arctanhz = 1 [ln(1+z) ln(1 z)], (48) [ ( arccothz = 1 ln 1+ 1 ) ( ln 1 1 )]. (49) z z 13

14 7 The principal values Arctanh and Arccoth Mathematica 8 defines the principal values of the inverse hyperbolic tangent and inverse hyperbolic cotangent, Arctanh and Arccoth, by employing the principal value of the complex logarithms in eqs. (48) and (49). We can define the principal value of the inverse hyperbolic tangent function by employing the principal value of the logarithm, Arctanhz = 1 [Ln(1+z) Ln(1 z)] (50) and Arccoth z = Arctanh ( ) [ ( 1 = 1 Ln 1+ 1 ) ( Ln 1 1 )] z z z Note that the branch points at z = ±1 are excluded from the above definitions, as Arctanh z and Arccoth z are divergent at these two points. The definition of the principal value of the inverse hyperbolic cotangent given in eq. (51) is deficient in one respect since it is not well-defined at z = 0. For this special case, Mathematica 8 defines Arccoth(0) = 1 iπ. (5) Of course, this discussion parallels that of Section. Moreover, alternative definitions of Arctanh z and Arccothz analogous to those defined in Appendix C for the corresponding inverse trigonometric functions can be found in ref. 4. There is no need to repeat the analysis of Section since a comparison of eqs. (1) and (13) with eqs. (50) and (51) shows that the inverse trigonometric and inverse hyperbolic tangent and cotangent functions are related by: (51) Arctanh z = iarctan( iz), (53) Arccothz = iarccot(iz). (54) Using these results, all other properties of the inverse hyperbolic tangent and cotangent functions can be easily derived from the properties of the corresponding arctangent and arccotangent functions. For example the branch cuts of these functions are easily obtained from eqs. (14) and (15). Arctanhz is discontinuous when z = x+iy crosses the branch cuts located on the real axis such that 5 y = 0 and < x < 1 and 1 < x <. (55) Arccothz is discontinuous when z = x + iy crosses the branch cuts located on the real axis such that y = 0 and 1 < x < 1. (56) 5 Note that for real w, we have tanhw 1 and cothw 1. Hence, for w = Arctanh z it is desirable to choose the branch cut to lie outside the interval on the real axis where Re z 1. Likewise, for w = Arccothz it is desirable to choose the branch cut to lie outside the interval on the real axis where Re z 1. 14

15 The relations between the single-valued and multi-valued functions can be summarized by: arctanhz = Arctanhz +inπ, n = 0, ±1, ±,, arccothz = Arccothz +inπ, n = 0, ±1, ±,. 8 The inverse hyperbolic functions: arcsinh and arccosh The inverse hyperbolic sine function is the solution to the equation: Letting v e w, we solve the equation z = sinhw = ew e w. v 1 v = z. Multiplying by v, one obtains a quadratic equation for v, The solution to eq. (57) is: v zv 1 = 0. (57) v = z +(1+z ) 1/. (58) Since z is a complex variable, (1+z ) 1/ is the complex square-root function. This is a multi-valued function with two possible values that differ by an overall minus sign. Hence, we do not explicitly write out the ± sign in eq. (58). To avoid ambiguity, we shall write v = z +(1+z ) 1/ = z +e 1 ln(1+z) = z +e 1 [Ln 1+z +iarg(1+z )] = z + 1+z 1/ e i arg(1+z). By definition, v e w, from which it follows that ( ) w = lnv = ln z + 1+z 1/ e i arg(1+z ). The solution to z = sinhw is w = arcsinhz. Hence, ( ) arcsinhz = ln z + 1+z 1/ e i arg(1+z ) (59) 15

16 The inverse hyperbolic cosine function is the solution to the equation: Letting v e w, we solve the equation z = cosw = ew +e w. v+ 1 v = z. Multiplying by v, one obtains a quadratic equation for v, The solution to eq. (60) is: v zv +1 = 0. (60) v = z +(z 1) 1/. Following the same steps as in the analysis of inverse hyperbolic sine function, we write w = arccoshz = lnv = ln [ z +(z 1) 1/], (61) where (z 1) 1/ is the multi-valued square root function. More explicitly, ( ) arccoshz = ln z + z 1 1/ e i arg(z 1) The multi-valued square root function satisfies: (z 1) 1/ = (z +1) 1/ (z 1) 1/. Hence, an equivalent form for the multi-valued inverse hyperbolic cosine function is: arccoshz = ln [ z +(z +1) 1/ (z 1) 1/], or equivalently, ) arccoshz = ln (z + z 1 1/ e i arg(z+1) e i arg(z 1). (6) Finally, we note the relations between the inverse trigonometric and the inverse hyperbolic functions: arcsinhz = iarcsin( iz), (63) arccoshz = ±i arccosz, (64) where the equalities in eqs. (63) and (64) are interpreted as set equalities for the multi-valued functions. The ± in eq. (64) indicates that both signs are employed in determining the members of the set of all possible arccoshz values. In deriving eq. (64), we have employed eqs. (6) and (61). In particular, the origin of the two possible signs in eq. (64) is a consequence of eq. (45) [and its hyperbolic analog, eq. (73)]. 16

17 9 The principal values Arcsinh and Arccosh The principal value of the inverse hyperbolic sine function, Arcsinh z, is defined by Mathematica 8 by replacing the complex logarithm and argument functions of eq. (59) by their principal values. That is, Arcsinhz = Ln (z + ) 1+z For the principal value of the inverse hyperbolic cosine function Arccoshz, Mathematica 8 chooses eq. (6) with the complex logarithm and argument functions replaced by their principal values. That is, (65) Arccoshz = Ln ( z + z +1 z 1 ) (66) In eqs. (65) and (66), the principal values of the square root functions are employed following the notation of eq. (3). The relation between the principal values of the inverse trigonometric and the inverse hyperbolic sine functions is given by Arcsinh z = iarcsin( iz), (67) as one might expect in light of eq. (63). A comparison of eqs. (39) and (66) reveals that { iarccosz, for either Im z > 0 or for Im z = 0 and Re z 1, Arccoshz = iarccosz, for either Im z < 0 or for Im z = 0 and Re z 1. (68) The existence of two possible signs in eq. (68) is not surprising in light of the ± that appears in eq. (64). Note that either choice of sign is valid in the case of Im z = 0 and Re z = 1, since for this special point, Arccosh(1) = Arccos(1) = 0. For a derivation of eq. (68), see Appendix F. The principal value of the inverse hyperbolic sine and cosine functions are singlevalued for all complex z. Moreover, due to the branch cut of the principal value square root function, 6 it follows that Arcsinh z is discontinuous when z = x + iy crosses lines on the imaginary axis such that x = 0 and < y < 1 and 1 < y <. (69) These two lines comprise the branch cuts of Arcsinh z, and each branch cut ends at a branch point located at z = i and z = i, respectively, due to the square root 6 One can check that the branch cut of the Ln function in eq. (65) is never encountered for any value of z. In particular, the branch cut of Ln can only be reachedif z+ 1+z is real and negative. But this never happens since if z + 1+z is real then z is also real. But for any real value of z, we have z + 1+z > 0. 17

18 function in eq. (65), although the square root function is not divergent at these points. The function Arcsinhz also possesses a branch point at the point of infinity, which can be verified by examining the behavior of Arcsinh(1/z) at the point z = 0. 7 The branch cut for Arccoshz derives from the standard branch cuts of the square root function and the branch cut of the complex logarithm. In particular, for real z satisfying z < 1, we have a branch cut due to (z+1) 1/ (z 1) 1/, whereas for real z satisfying < z 1, the branch cut of the complex logarithm takes over. Hence, it follows that Arccoshz is discontinuous when z = x + iy crosses lines on the real axis such that 8 y = 0 and < x < 1. (70) In particular, there are branch points at z = ±1 due to the square root functions in eq.(66)andabranchpointatthepointofinfinityduetothelogarithm[cf.footnote5]. As a result, eq. (70) actually represents two branch cuts made up of a branch cut from z = 1 to z = 1 followed by a second branch cut from z = 1 to the point of infinity. 9 The relations between the single-valued and multi-valued functions can be obtained by following the same steps used to derive eqs. (43) and (44). Alternatively, we can make use of these results along with those of eqs. (63), (64), (67) and (68). The end result is: arcsinhz = ( 1) n Arcsinhz +inπ, n = 0, ±1, ±,, (71) arccoshz = ±Arccoshz +inπ, n = 0, ±1, ±,, (7) where either ±Arccoshz can be employed to obtain a possible value of arccoshz. In particular, the choice of n = 0 in eq. (7) implies that: arccoshz = arccoshz, (73) which should be interpreted as a set equality. This completes our survey of the multi-valued complex inverse trigonometric and hyperbolic functions and their single-valued principal values. 7 In the complex plane, the behavior of the complex function F(z) at the point of infinity, z =, corresponds to the behavior of F(1/z) at the origin of the complex plane, z = 0 [cf. footnote 3]. Since the argument of the complex number 0 is undefined, the argument of the point of infinity is likewise undefined. This means that the point of infinity (sometimes called complex infinity) actually corresponds to z =, independently of the direction in which infinity is approached in the complex plane. Geometrically, the complex plane plus the point of infinity can be mapped onto a surface of a sphere by stereographic projection. Place the sphere on top of the complex plane such that the origin of the complex plane coincides with the south pole. Consider a straight line from any complex number in the complex plane to the north pole. Before it reaches the north pole, this line intersects the surface of the sphere at a unique point. Thus, every complex number in the complex plane is uniquely associated with a point on the surface of the sphere. In particular, the north pole itself corresponds to complex infinity. For further details, see Chapter 5 of ref Note that for real w, we have coshw 1. Hence, for w = Arccoshz it is desirable to choose the branch cut to lie outside the interval on the real axis where Re z 1. 9 Given that the branch cuts of Arccoshz and iarccosz are different, it is not surprising that the relation Arccoshz = iarccosz cannot be respected for all complex numbers z. 18

19 APPENDIX A: The argument of a complex number In this Appendix, we examine the argument of a non-zero complex number z. Any complex number can be written as, we write: z = x+iy = r(cosθ+isinθ) = re iθ, (A.1) where x = Re z and y = Im z are real numbers. The complex conjugate of z, denoted by z, is given by z = x iy = re iθ. The argument of z is denoted by θ, which is measured in radians. However, there is an ambiguity in definition of the argument. The problem is that sin(θ +π) = sinθ, cos(θ+π) = cosθ, since the sine and the cosine are periodic functions of θ with period π. Thus θ is defined only up to an additive integer multiple of π. It is common practice to establish a convention in which θ is defined to lie within an interval of length π. This defines the so-called principal value of the argument, which we denote by θ Arg z (note the capital A). The most common convention, which we adopt in these notes, is to define the single-valued function Arg z such that: π < Arg z π. (A.) In many applications, it is convenient to define a multi-valued argument function, arg z Arg z +πn = θ +πn, n = 0,±1,±,±3,.... (A.3) This is a multi-valued function because for a given complex number z, the number arg z represents an infinite number of possible values. A.1. The principal value of the argument function Any non-zero complex number z can be written in polar form z = z e iargz, (A.4) where argz is a multi-valued function defined in eq. (A.3) It is convenient to have an explicit formula for Arg z in terms of argz. First, we introduce some notation: [x] means the largest integer less than or equal to the real number x. That is, [x] is the unique integer that satisfies the inequality x 1 < [x] x, for real x and integer [x]. (A.5) 19

20 For example, [1.5] = [1] = 1 and [ 0.5] = 1. With this notation, one can write Arg z in terms of argz as follows: [ 1 Arg z = argz +π argz ], (A.6) π where [ ] denotes the bracket (or greatest integer) function introduced above. It is straightforward to check that Arg z as defined by eq. (A.6) does indeed fall inside the principal interval, π < θ π. A more useful equation for Arg z can be obtained as follows. Using the polar representation of z = x + iy given in eq. (A.1), it follows that x = rcosθ and y = rsinθ. From these two results, one easily derives, z = r = x +y, tanθ = y x. (A.7) We identify θ = Arg z in the convention where π < θ π. In light of eq. (A.7), it is tempting to identify Arg z with arctan(y/x). However, the real function arctan x is a multi-valued function for real values of x. It is conventional to introduce a singlevalued real arctangent function, called the principal value of the arctangent, which is denoted by Arctan x and satisfies 1π Arctan x 1 π [cf. eq. (18)]. Since π < Arg z π, it follows that Arg z cannot be identified with arctan(y/x) in all regions of the complex plane. The correct relation between these two quantities is easily ascertained by considering the four quadrants of the complex plane separately. The quadrants of the complex plane (called regions I, II, III and IV) are illustrated in the figure below: y II I x III IV The principal value of the argument of z = x + iy in terms of its real part x and imaginary part y is given in Table 1, assuming that z lies within one of the four quadrants of the complex plane. Note that Arg z = Arctan(y/x) is valid only in quadrants I and IV. If z lies within quadrants II or III, one must add or subtract π in order to ensure that 1π < Arg z < π or π < Arg z < 1 π, respectively For finite non-zero values of y/x, the principal value of the arctangent function lies inside the interval0 < Arctan(y/x) < 1 π ify/x > 0and 1 π < Arctan(y/x) < 0ify/x < 0. Forcompleteness, we note that 0, if y = 0 and x 0, 1 Arctan(y/x) = π, if x = 0 and y > 0, 1 π, if x = 0 and y < 0, undefined, if x = y = 0. 0

21 Table 1: Formulae for the argument of a complex number z = x+iy. Quadrant Sign of x and y Arg z I x > 0, y > 0 Arctan(y/x) II x < 0, y > 0 π +Arctan(y/x) III x < 0, y < 0 π +Arctan(y/x) IV x > 0, y < 0 Arctan(y/x) Table: Formulaefortheargumentofacomplexnumber z = x+iy whenz isrealorpure imaginary. By convention, the principal value of the argument satisfies π < Arg z π. Quadrant border type of complex number z Conditions on x and y Arg z IV/I real and positive x > 0, y = 0 0 I/II pure imaginary with Im z > 0 x = 0, y > 0 1 π II/III real and negative x < 0, y = 0 π III/IV pure imaginary with Im z < 0 x = 0, y < 0 1 π origin zero x = y = 0 undefined Cases where z lies on the border between two adjacent quadrants are considered separately in Table. In particular, note that the argument of zero is undefined. Since z = 0 if and only if z = 0, eq. (A.4) remains valid despite the fact that arg0 is not defined. When studying the properties of arg z and Argz below, we shall always assume implicitly that z 0. A.. Properties of the multi-valued argument function We can view a multi-valued function f(z) evaluated at z as a set of values, where each element of the set corresponds to a different choice of some integer n. For example, given the multi-valued function argz whose principal value is Arg z θ, then argz consists of the set of values: argz = {θ, θ+π, θ π, θ+4π, θ 4π, }. (A.8) Given two multi-valued functions, e.g., f(z) = F(z) + πn and g(z) = G(z) + πn, where F(z) and G(z) are the principal values of f(z) and g(z) respectively, then 1

22 f(z) = g(z) if and only if for each point z, the corresponding set of values of f(z) and g(z) precisely coincide: {F(z), F(z)+π, F(z) π, } = {G(z), G(z)+π, G(z) π, }. (A.9) Sometimes, one refers to the equation f(z) = g(z) as a set equality since all the elements of the two sets in eq. (A.9) must coincide. We add two additional rules to the concept of set equality. First, the ordering of terms within the set is unimportant. Second, we only care about the distinct elements of each set. That is, if our list of set elements has repeated entries, we omit all duplicate elements. To see how the set equality of two multi-valued functions works, let us consider the multi-valued function arg z. One can prove that: arg(z 1 z ) = argz 1 +argz, (A.10) ( ) z1 arg = argz 1 argz. (A.11) arg z ( ) 1 z = argz = argz. (A.1) To prove eq. (A.10), consider z 1 = z 1 e iarg z 1 and z = z e iarg z. The arguments of these two complex numbers are: argz 1 = Arg z 1 +πn 1 and argz = Arg z +πn, where n 1 and n are arbitrary integers. [One can also write argz 1 and argz in set notationasineq.(a.8).] Thus, onecanalsowritez 1 = z 1 e iarg z 1 andz = z e iarg z, since e πin = 1 for any integer n. It then follows that z 1 z = z 1 z e i(arg z 1+Arg z ), where we have used z 1 z = z 1 z. Thus, arg(z 1 z ) = Arg z 1 + Arg z + πn 1, where n 1 is also an arbitrary integer. Therefore, we have established that: argz 1 +argz = Arg z 1 +Arg z +π(n 1 +n ), arg(z 1 z ) = Arg z 1 +Arg z +πn 1, where n 1, n and n 1 are arbitrary integers. Thus, argz 1 + argz and arg(z 1 z ) coincide as sets, and so eq. (A.10) is confirmed. One can easily prove eqs. (A.11) and (A.1) by a similar method. In particular, if one writes z = z e iargz and employs the definition of the complex conjugate (which yields z = z and z = z e iargz ), then it follows that arg(1/z) = argz = argz. As an instructive example, consider the last relation in the case of z = 1. It then follows that arg( 1) = arg( 1), as a set equality. This is not paradoxical, since the sets, arg( 1) = {±π, ±3π, ±5π,...} and arg( 1) = { π, 3π, 5π,...},

23 coincide, as they possess precisely the same list of elements. Now, for a little surprise: argz argz. (A.13) To see why this statement is surprising, consider the following false proof. Use eq. (A.10) with z 1 = z = z to derive: argz = argz +argz? = argz, [FALSE!!]. (A.14)? The false step is the one indicated by the symbol = above. Given z = z e iargz, one finds that z = z e i(arg z+πn) = z e iarg z, and so the possible values of arg(z ) are: arg(z ) = {Arg z, Arg z +π, Arg z π, Arg z +4π, Arg z 4π, }, whereas the possible values of argz are: argz = {Arg z, (Arg z +π), (Arg z π), (Arg z +4π), } = {Arg z, Arg z +4π, Arg z 4π, Arg z +8π, Arg z 8π, }. Thus, argz is a subset of arg(z ), but half the elements of arg(z ) are missing from argz. These are therefore unequal sets, as indicated by eq. (A.13). Now, you should be able to see what is wrong with the statement: argz +argz? = argz. (A.15) When you add arg z as a set to itself, the element you choose from the first argz need not be the same as the element you choose from the second argz. In contrast, argz means take the set argz and multiply each element by two. The end result is that argz contains only half the elements of argz+argz as shown above. Similarly, for any non-negative integer n =,3,4,..., argz n = argz +argz + argz n argz. }{{} n (A.16) In light of eq. (A.1), if we replace z with z above, we obtain the following generalization of eq. (A.16), argz n n argz, for any integer n 0, ±1. (A.17) Here is one more example of an incorrect proof. Consider eq. (A.11) with z 1 = z z. Then, you might be tempted to write: ( z arg = arg(1) = argz argz = z)? 0. 3

24 This is clearly wrong since arg(1) = πn, where n is the set of integers. Again, the error occurs with the step: argz argz? = 0. (A.18) The fallacy of this statement is the same as above. When you subtract argz as a set from itself, the element you choose from the first argz need not be the same as the element you choose from the second argz. A.3. Properties of the principal value of the argument The properties of the principal value Arg z are not as simple as those given in eqs. (A.10) (A.1), since the range of Arg z is restricted to lie within the principal range π < Arg z π. Instead, the following relations are satisfied: Arg (z 1 z ) = Arg z 1 +Arg z +πn +, Arg (z 1 /z ) = Arg z 1 Arg z +πn, where the integers N ± are determined as follows: 1, if Arg z 1 ±Arg z > π, N ± = 0, if π < Arg z 1 ±Arg z π, 1, if Arg z 1 ±Arg z π. If we set z 1 = 1 in eq. (A.0), we find that { Argz, if Im z = 0 and z 0, Arg(1/z) = Arg z = Argz, if Im z 0. (A.19) (A.0) (A.1) (A.) Note that for z real, both 1/z and z are also real so that in this case z = z and Arg(1/z) = Arg z = Arg z. Finally, Arg(z n ) = narg z +πn n, (A.3) where the integer N n is given by: N n = [ 1 n ] π Arg z, (A.4) and [ ] is the greatest integer bracket function introduced in eq. (A.5). It is straightforward to verify eqs. (A.19) (A.) and eq. (A.3). These formulae follow from the corresponding properties of arg z, taking into account the requirement that Arg z must lie within the principal interval, π < θ π. 4

25 APPENDIX B: The complex logarithm and its principal value B.1. Definition of the complex logarithm In order to define the complex logarithm, one must solve the complex equation: z = e w, (B.1) for w, where z is any non-zero complex number. If we write w = u+iv, then eq. (B.1) can be written as e u e iv = z e iargz. (B.) Eq. (B.) implies that: z = e u, v = argz. The equation z = e u is a real equation, so we can write u = ln z, where ln z is the ordinary logarithm evaluated with positive real number arguments. Thus, w = u+iv = ln z +iargz = ln z +i(arg z +πn), n = 0, ±1, ±, ±3,... (B.3) We call w the complex logarithm and write w = lnz. This is a somewhat awkward notation since in eq. (B.3) we have already used the symbol ln for the real logarithm. We shall finesse this notational quandary by denoting the real logarithm in eq. (B.3) by the symbol Ln. That is, Ln z shall denote the ordinary real logarithm of z. With this notational convention, we rewrite eq. (B.3) as: lnz = Ln z +iargz = Ln z +i(arg z +πn), n = 0, ±1, ±, ±3,... (B.4) for any non-zero complex number z. Clearly, lnz is a multi-valued function (as its value depends on the integer n). It is useful to define a single-valued function complex function, Ln z, called the principal value of lnz as follows: Lnz = Ln z +iarg z, π < Arg z π, (B.5) which extends the definition of Ln z to the entire complex plane (excluding the origin, z = 0, where the logarithmic function is singular). In particular, eq. (B.5) implies that Ln( 1) = iπ. Note that for real positive z, we have Arg z = 0, so that eq. (B.5) simply reduces to the usual real logarithmic function in this limit. The relation between lnz and its principal value is simple: lnz = Ln z +πin, n = 0, ±1, ±, ±3,.... 5

26 B.. Properties of the real and complex logarithm The properties of the real logarithmic function (whose argument is a positive real number) are well known: e lnx = x, ln(e a ) = a, (B.6) (B.7) ln(xy) = ln(x)+ln(y), (B.8) ( ) x ln = ln(x) ln(y), y (B.9) ( ) 1 ln = ln(x), x (B.10) for positive real numbers x and y and arbitrary real number a. Repeated use of eqs. (B.8) and (B.9) then yields lnx n = nlnx, for arbitrary integer n. (B.11) We now consider which of the properties given in eqs. (B.6) (B.11) apply to the complex logarithm. Since we have defined the multi-value function ln z and the singlevalued function Ln z, we should examine the properties of both these functions. We begin with the multi-valued function ln z. First, we examine eq. (B.6). Using eq. (B.4), it follows that: e lnz = e Ln z e iarg z e πin = z e iarg z = z. (B.1) Thus, eq. (B.6) is satisfied. Next, we examine eq. (B.7) for z = x+iy: ln(e z ) = Ln e z +i(arg e z ) = Ln(e x )+i(y +πk) = x+iy +πik = z +πik, wherekisanarbitraryinteger. Inderivingthisresult, weusedthefactthate z = e x e iy, which implies that arg(e z ) = y +πk. 11 Thus, ln(e z ) = z +πik z, unless k = 0. (B.13) This is not surprising, since ln(e z ) is a multi-valued function, which cannot be equal to the single-valued function z. Indeed eq. (B.7) is false for the multi-valued complex logarithm. As a check, let us compute ln(e lnz ) in two different ways. First, using eq. (B.1), it follows that ln(e lnz ) = lnz. Second, using eq. (B.13), ln(e lnz ) = lnz +πik. This 11 Note that Arg e z = y + πn, where N is chosen such that π < y + πn π. Moreover, eq. (A.3) implies that arge z = Arg e z +πn, where n = 0,±1,±,... Hence, arg(e z ) = y +πk, where k = n+n is still some integer. 6

27 seems to imply that lnz = lnz+πik. In fact, the latter is completely valid as a set equality in light of eq. (B.4). We now consider the properties exhibited in eqs. (B.8) (B.10). Using the definition of the multi-valued complex logarithms and the properties of arg z given in eqs. (A.10) (A.1), it follows that eqs. (B.8) (B.11) are satisfied as set equalities: ln(z 1 z ) = lnz 1 +lnz, (B.14) ( ) z1 ln = lnz 1 lnz. (B.15) ln z ( ) 1 z = lnz. (B.16) However, one must be careful in employing these results. One should not make the mistake of writing, for example, lnz + lnz? = lnz or lnz lnz? = 0. Both these latter statements are false for the same reasons that eqs. (A.14) and (A.18) are not identities under set equality. In general, the multi-valued complex logarithm does not satisfy eq. (B.11) as a set equality. In particular, lnz n nlnz, for integer n 0, ±1, (B.17) as a consequence of eq. (A.17). We next examine the properties of the single-valued function Ln z. Again, we examine the six properties given by eqs. (B.6) (B.11). First, eq. (B.6) is trivially satisfied since e Ln z = e Ln z e iarg z = z e iarg z = z. (B.18) However, eq. (B.7) is generally false. In particular, for z = x+iy Ln(e z ) = Ln e z +i(arg e z ) = Ln(e x )+i(arg e iy ) = x+iarg (e iy ) [ ] [ 1 = x+iarg(e iy )+πi arg(eiy ) 1 = x+iy +πi π y ] π [ 1 = z +πi Im z ], (B.19) π after using eq. (A.6), where [ ] is the greatest integer bracket function defined in eq. (A.5). Thus, eq. (B.7) is satisfied only when π < y π. For values of y outside the principal interval, eq. (B.7) contains an additive correction term as shown in eq. (B.19). As a check, let us compute Ln(e Lnz ) in two different ways. First, using eq. (B.18), it follows that Ln(e Ln z ) = Ln z. Second, using eq. (B.19), Ln(e Ln z ) = lnz +πi [ 1 Im Ln z π ] = Ln z +πi 7 [ 1 Arg z π ] = Ln z,

28 where we have used Im Ln z = Arg z [see eq. (B.5)]. In the last step, we noted that 0 1 Arg(z) π < 1, due to eq. (A.), which implies that the integer part of 1 Arg z/(π) is zero. Thus, the two computations agree. We now consider the properties exhibited in eqs. (B.8) (B.11). Ln z may not satisfy any of these properties due to the fact that the principal value of the complex logarithmmust lie inthe interval π < ImLnz π. Using theresults of eqs. (A.19) (A.4), it follows that Ln (z 1 z ) = Ln z 1 +Ln z +πin +, Ln (z 1 /z ) = Ln z 1 Ln z +πin, (B.0) (B.1) Ln(z n ) = nln z +πin n (integer n), (B.) where the integers N ± = 1, 0 or +1 and N n are determined by eqs. (A.1) and (A.4), respectively, and { Ln(z)+πi, if z is real and negative, Ln(1/z) = (B.3) Ln(z), otherwise. Note that eq. (B.8) is satisfied if Re z 1 > 0 and Re z > 0 (in which case N + = 0). In other cases, N + 0 and eq. (B.8) fails. Similar considerations also apply to eqs. (B.9) and(b.10). Forexample, eq. (B.10)issatisfiedbyLn z unless Arg z = π (equivalently for negative real values of z), as indicated by eq. (B.3). In particular, one may use eq. (B.) to verify that: Ln[( 1) 1 ] = Ln( 1)+πi = πi+πi = πi = Ln( 1), as expected, since ( 1) 1 = 1. B.3. Properties of the generalized power function The generalized complex power function is defined via the following equation: w = z c = e clnz, z 0. (B.4) Having defined the multi-valued complex power function, we are now able to compute ln(z c ) for arbitrary complex number c and complex variable z, ln(z c ) = ln(e c lnz ) = ln(e c(ln z+πim) ) = ln(e cln z e πimc ) = ln(e cln z )+ln(e πimc ) = c(lnz +πim)+πik ( = c lnz +πik = c lnz + πik ), (B.5) c 8