Centrum voor Wiskunde en Informatica REPORTRAPPORT Asymptotics of zeros of incomplete gamma functions N.M. Temme Department of Analysis, Algebra and Geometry AM-R9402 994
Asymptotics of Zeros of Incomplete Gamma Functions N. M. Temme CWI, P.O. Box 94079, 090 GB Amsterdam, The Netherlands e-mail: nicot@cwi.nl This paper is dedicated to Luigi Gatteschi, in admiration for his excellent contributions to the study of zeros of orthogonal polynomials. Abstract We consider the complex zeros with respect to z of the incomplete gamma functions (a; z) and?(a; z), with a real and positive. In particular we are interested in the case that a is large. The zeros are obtained from approximations that are computed by using uniform asymptotic expansions of the incomplete gamma functions. The complex zeros of the complementary error function are used as a rst approximations. Applications are discussed for the zeros of the partial sums s n (z) = P n j=0 zj =j! of exp(z). 99 Mathematics Subject Classication: 33B5, 33B20, 4A60, 65U05. Keywords & Phrases: incomplete gamma functions, zeros of incomplete gamma functions, error function, uniform asymptotic expansion.. Introduction The incomplete gamma functions are dened by P (a; z) =?(a) Z z 0 t a? e?t dt; Q(a; z) =?(a) Z z t a? e?t dt; (:) with P (a; z) + Q(a; z) = : (:2) We take a > 0 and z complex, j arg zj <. The function (a; z) = z?a P (a; z) is an entire function of both parameters a and z. When a is a non-negative integer the incomplete gamma functions are very simple: (n + ; z) = n! [? e?z s n (z)]?(n + ; z) = n! e?z s n (z): n = 0; ; 2; : : :; (:3)
2 in which s n (z) is the rst part of the Taylor series of the exponential function: s n (z) = nx j=0 z j j! ; n = 0; ; 2; : : : : (:4) We consider the zeros with respect to z of the incomplete gamma functions, with a xed and positive. The method for obtaining the zeros is based on asymptotic expansions of the incomplete gamma functions, that hold for large values of a and are uniformly valid with respect to z. We derive asymptotic expansions of the zeros. The expansions are of the same kind as the expansion of the positive solution z(a; q) of the inversion problem?(a; z) = q with q 2 (0; ); see Temme (992). This inversion problem is of importance in probability theory and mathematical statistics. In Kolbig (972) the zeros of the incomplete gamma function (a; z) are considered from a dierent point of view. That paper gives in particular information on the zeros with respect to a, with z real and positive. It gives curves of the zeros of (xw; x) in the w?plane; the trajectories are parameterised by the positive parameter x. Kolbig's paper contains many references to earlier papers. The asymptotic distribution of zeros of the partial sums s n (z) of the exponential function (see (.3), (.4)) has received much attention in the literature. For a recent overview we refer to Varga (990). 2. Uniform asymptotic expansions of P and Q We obtain an asymptotic expansion of the zeros of the incomplete gamma functions using uniform asymptotic expansion of these functions as given in Temme (979). First we summarize these results. The incomplete gamma functions have the following representations p P (a; z) =? erfc a=2? R 2 a (); p Q(a; z) = erfc a=2 + R a (); 2 erfc is the error function dened by The real parameter in (2.) is dened by (2:) Z erfc z = p 2 e?t2 dt: (2:2) z 2 2 =?? ln ; = z=a; sign() = sign(? ): (2:3) The condition on the sign of holds for positive values of z. For complex values we dene the branch of the square root by analytic continuation. Alternatively, = (? ) s 2?? ln (? ) 2 ; = z=a; (2:4)
3 where the square root takes the value when =. For the function R a () we derived an asymptotic expansion. Writing R a () = e? 2 a 2 p S a (); (2:5) 2a we have S a () X n=0 C n () ; as a! : (2:6) n a This expansion holds in an unbounded domain of the parameter 2 IR; in particular the expansion holds in a large neighbourhood of the point = 0, which corresponds with z = a, a turning point in the asymptotic behaviour of the incomplete gamma functions for large values of the parameter a. When a is positive, the expansion (2.6) holds uniformly with respect to z in the complex plane. The rst two coecients in (2.6) are C 0 () =?? ; C () = 3? (? ) 3? (? ) 2? 2(? ) : (2:7) These two (and all higher coecients) have a removable singularity at = 0 ( = ; z = a). All C n () are analytic at = 0. The higher coecients can be obtained from the recursion C n () = d d C n?() +? n; n ; (2:8) where the numbers n appear in the well-known asymptotic expansion of the Euler gamma function. That is,? (a) X n=0 (?) n n a?n ;? (a) X n=0 n a?n ; a! ; (2:9) where The rst few n are? (a) = r a 2 ea a?a?(a); a > 0: (2:0) 0 = ; =? 2 ; 2 = 288 ; 3 = 39 5840 : 3. Asymptotic expansions of the zeros of the incomplete gamma functions We concentrate on the function Q(a; z). We rst compute the zeros in terms of the parameter, by using the representation in (2.), with large values of a. Afterwards we compute and z from (2.3).
4 We rst repeat the steps in the inversion problem 2 erfc p a=2 + R a () = q; (3:) which has been considered in Temme (992) when q 2 [0; ]. The procedure started with the solution 0 of the equation erfc 0 pa=2 = q: (3:2) 2 and the solution dened by (3.) is written as (q; a) = 0 (q; a) + "( 0 ; a). The quantity "( 0 ; a) is written in the form of the expansion "( 0 ; a) " a + " 2 a 2 + " 3 a 3 + : : : ; (3:3) as a!. The coecients " i can be written explicitly as functions of 0. In [4] a dierential equation for " was derived from which the expansion (3.3) could be obtained by perturbation methods. The same dierential equation can be used in the present problem to determine the zeros. Observe that (3.) yields the relation dq = d d d Q(a; z) = d dz Q(a; z) dz d : Using (.) and (2.3), one obtains after straightforward calculations where? (a) is dened in (2.0), and dq =? d? (a) r a 2 f()e? 2 a 2 ; (3:4) f() =? ; (3:5) the relation between and being given in (2.3). For small values of we can expand From (3.2) it follows that f() =? 3 + 2 2 + : : : : (3:6) r dq a =? d 0 2 e? a 2 2 0 : (3:7) Upon dividing the two dierential equations in (3.4), (3.7) the quantity q is eliminated. That is, d =? (a) d 0 f() e a( 2 2?0 2 ) : (3:8) Replacing with 0 + ", one has f( 0 + ") + d" d 0 =? (a)e a"( 0+ 2 ") ; a relation between " and 0, with a as (large) parameter.
As remarked earlier, we use this dierential equation also in the present case. The value q = 0 gives in our earlier paper the trivial solution 0 =, i.e., z = +; this solution is not considered in the present case. Obviously, the above approach does not require the condition q 2 [0; ]. In fact q may assume any complex value; when we take q = 0 the quantity 0 is associated with the nite zeros of the complementary error function in (2.). More information on these zeros will be given in the following section. The quantities " i introduced in (3.3) can be computed as in [4]. Let 0 be a nite complex number satisfying (3.2), with a a given positive number. Then 0 will be a rst order approximation (for large values of a) of, that solves the equation 2 erfc p a=2 + R a () = q: (3:9) This equation is equivalent with Q(a; a) = 0, the relation between and being given in (2.4). Higher order approximations to are obtained by writing 0 + " a + " 2 a 2 + " 3 a 3 + : : : : (3:0) 5 The rst few " i are given by " = ln f(); 2 3 " 2 = + 2? 2f 2? 2f? 2f 2 "? 2f 2 "? 2? 6 2 " 2 ; 2 5 " 3 =? 30 + 2f 2 " + 2f 2 " + 24f 2 3 " + 6" 3 3? 2f 2 + 60f 3 2 " + 3f 2 2 + 72f 3 + 42f 4 + 8f 3 3 " 2 + 6f 2 4 " 2 + 36f 4 " + 2" 2 3 f + 2" 2 2 f 2? 2" + 3 " + f 3? 2f + 2" 2 2 f 4 : Here, the quantity equals 0 and f is the function f(), which is dened in (3.5). For small values of 0 it is convenient to have series expansions of the coecients " i (note that 0 may be quite small, when a is large). From [4] we have " =? 3 + 36 + 620 2? 7 6480 3 + 5 844 4? 382725 5? 0 6329600 6 + : : : ; " 2 =? 7 405? 7 2592 + 533 20420 2? 579 2099520 3 + 09 749600 4 + 027 25942400 5 + : : : ; " 3 = + 449 02060? 6349 20995200 + 29233 3674600 2 + 346793 5290790400 3 + : : : ; again with replaced with 0. When we have obtained from (3.0), we can compute by inverting the relation between and in (2.4). For small values of we can use a series expansion of in terms of. Inverting 2 2 = 2 (? )2? 3 (? )3 + 4 (? )4 + : : : we obtain = + + 3 2 + 36 3? 270 4 + 4320 5 + : : : :
6 4. More information on the zeros and numerical examples From Fettis, Caslin & Cramer (973) we know that two innite strings of zeros of erfc z occur in the neighbourhood of the diagonals y = x in the left half plane x < 0, z = x + iy. The rst few zeros are given in Table 4.. Numerical values of the rst 00 zeros of erfcz and asymptotic approximations of the zeros are also given by Fettis et al. A rst order approximation reads z (? i)p (k? =8) : (4:) k k x k i y k -.3548... i.9946... 2-2.7704... i 2.694... 3-2.78438... i 3.23533... 4-3.2874... i 3.69730... 5-3.72594... i 4.060... Table 4.. First ve pairs z k = x k i y k of zeros of erfc z When is a zero of equation (3.9) and we use (3.0) as an asymptotic expansion of for large positive values of a, we nd that the?zeros of Q(a; a) in terms of the parameter occur in the neighbourhood of the diagonals < = =, with < < 0. It is therefore of interest to know the original?contours of these diagonals under the mapping (2.3). To study the mapping and the pre-images of the diagonals, let us write = + i and = exp(i). Then the relation between and given in (2.3) can be written in terms of the real equations 2 (2? 2 ) = cos?? ln ; (4:2) = sin? : On the diagonals = the rst equation becomes cos = + ln. In terms of cartesian coordinates ( = x + iy) we have x 2 + y 2 = e 2(x?) : (4:3) This equation denes an almond-shaped closed curve between x = and x =?0:278 : : :, the solution of the equation?x = exp(x? ); see Figure 4.. In [6] this curve is called the Szego curve and dened as D = n 2 C : e? = and jj o : As discussed in detail in [6], when a = n (integer) the n?zeros of?(n; n) (a polynomial in, see the second line in (.3)) approach this curve when a!. The equation in (4.3) denes also solutions for x > ; there are two branches starting in x = ; y = 0 and extending to innity along the curves y = exp(x? ). These branches have no meaning in connection with the zeros of?(a; a) when a is an integer. However, when a is not an integer?(a; a) has an innity of?zeros of which about [a] zeros are located near D with phases in the interval (?; ). An innite number of zeros is located near the branches
of equation (4.3) with x >, but the phases of the zeros are in the intervals (?2;?) and (; 2). To see this we give more details on the mapping given in (4.2). Finite singular points of the mapping 7! () can be found by considering the derivative d=d = (? )=(). The point = is a regular point, but = exp(2in)(n = ; 2; : : :) are points where the derivative d=d vanishes. They correspond to the?values satisfying 2 2 n =?2in, giving the singular points = 2 p exp(3i=4) for n =. The points are located at the diagonals in the left half plane, and they correspond to = exp(2i). It can be proved (details will not be given here) that the?sector j arg j < 2; ( 6= 0) is mapped conformally and one-to-one onto the?plane cut along two branch lines, which are parts of the hyperbolas = 2 with? p 2. The boundaries of this sector, that is, the half lines arg = 2i are mapped onto these branch cuts. For more details we refer to [3]. 7 A = O B < -C Figure 4..?zeros of?(a; a) with a = 30:, along curves dened by x 2 + y 2 = exp[2(x? )]; shown are the zeros with phases in the interval [0; 2]. There is a conjugate set of zeros with phases in the interval [?2; 0]. To understand the role of the diagonals in the left half?plane in connection with the location of the zeros of the incomplete gamma function?(a; a) in the?plane, we concentrate on a few points O; A; B; C on the diagonal =?; 0 (because of the symmetry it is sucient to consider 0). Let O denote the origin, A the point where =?, B the point where =?2, and C a point where <?2. Then the segment [O; A] corresponds in the?plane to the upper part of the almond-shaped curve in Figure 4., [A; B] corresponds to the lower part of this curve (the phase of is now in the interval [; 2]), and [B; C] with part of the branch extending to the right of =, with = < 0 (and with the phase of still in [; 2]). This description of the mapping with respect to the diagonals in the left?plane makes clear how the approximations of the zeros of?(a; a) in the?plane are mapped onto the?plane. The?zeros in the neighbourhood of the segment [O; A] are mapped onto the upper part of the almond-shaped curve in the?plane. From the estimates given for the zeros of
8 the error function, see (4.), we see that there are about a=2 zeros along this curve. Another a=2 are located along the lower part of the almond (with phases belonging to [; 2]); they correspond to?values in the neighbourhood of the segment [A; B]. When a = n (integer) there are exactly n zeros along the complete almond. In that case it is not needed to prescribe the phases of the zeros on the lower part of the almond, since then?(a; a) is not multi-valued. The zeros corresponding to?zeros along the diagonal = in the third quadrant of the?plane yield a conjugate set of zeros in the?plane. The diagonals in the right half plane 0 correspond to the branches in the? plane that start in the point = and extend to i along the curves y = exp(x? ), with phases of belonging to [0; ]. Along these 2 contours in the? plane we nd zeros of (a; a). This follows from the minus sign in the error function representation of the function P (a; z) in (2.). In a separate publication (a rst orientation on the computation of incomplete gamma functions for complex parameters is given in [5]) we will verify numerically the asymptotic expansions of the zeros of?(a; a) and (a; a) derived in this paper. In Figure 4. we show the rst 50 zeros of?(a; a) with a = 30: with phases in the interval (0; 2). In the upper half plane the phases of the zeros are in the interval (0; ), in the lower half plane they are in the interval (; 2). A similar picture can be given for zeros with phase in (?2; 0). When a is an integer the function?(a; a) is single-valued, and then we can consider phases just in the interval (?; ). In that case the zeros along the branches extending to innity do not occur. Bibliography [] H.E. Fettis, J.C. Caslin and K.R. Cramer (973), Complex zeros of the error function and of the complementary error function, Math. Comp. 27 40{407. [2] K.S. Kolbig (972), On the zeros of the incomplete gamma function, Math. Comp. 26 75{755 [3] N.M. Temme (979), The asymptotic expansions of the incomplete gamma functions, SIAM J. Math. Anal. 0 757-766. [4] N.M. Temme (992), Asymptotic inversion of incomplete gamma functions, Math. Comp. 58 755-764. [5] N.M. Temme (994), Computational aspects of incomplete gamma functions with large complex parameters. Submitted for publication. [6] R.S. Varga (990), Scientic computation on mathematical problems and conjectures, CBMS-NSF Regional Conference Series in Applied Mathematics 60, SIAM.