Application of the Argument Principle to Functions Expressed as Mellin Transforms
aa r X i v : . [ m a t h . G M ] J a n Application of the Argument Principle to FunctionsExpressed as Mellin Transforms
Bjoern S. Schmekel ∗ Department of Physics, College of Studies for Foreign DiplomaRecipients at the University of Hamburg, 20355 Hamburg, Germany
January 15, 2021
Abstract
We describe a numerical algorithm for evaluating the numbers of roots minus the number ofpoles contained in a region based on the argument principle with the function of interest beingwritten as a Mellin transformation of a usually simpler function. Because the function to betransformed may be simpler than its Mellin transform whose roots are to be sought we expressthe final integrals in terms of the former accepting higher dimensional integrals. Nonlinearterms are expressed as convolutions approximating reciprocal values by exponential sums. Asan example the final expression is applied to the Riemann Zeta function. The procedure isvery inefficient numerically. However, depending on the function to be investigated it may bepossible to find analytical estimates of the resulting integrals.
Keywords—
Root-Finding Algorithms, Argument Principle, Mellin Transformation, Riemann ZetaFunction
Object of this paper is to compute the number of roots minus the number of poles enclosed by a closedcontour C using the argument principle N R − N P = 12 πi I C f ′ ( s ) f ( s ) ds (1.1)where f ( z ) is a meromorphic function on and inside of the contour C which can be represented as anauxilliary function multiplied by the Mellin transform of another function f ( s ) = K ( s ) Z ( s ) = K ( s ) Z ∞ z ( t ) t s − dt (1.2)assuming the Melling transform exists on and inside of the contour C . The latter is chosen such that itdoes not run over any poles or roots. We are interested in cases where z ( t ) is a simple function thereforeexpressing the final result in terms of K ( s ) as well as z ( t ) instead of Z ( s ). Furthermore, we are lookingfor an expression such that eqn. 1.1 can be expressed as a multi-dimensional integral of z ( t ). Obviously,the integrand in eqn. 1.1 is a nonlinear functional of z ( t ) (and K ( s ) ), so arriving at such a result is notcompletely straightforward. In a first step we deal with the nonlinearities by approximating the reciprocalvalue of f ( s ) in the argument principle by an exponential sum [1, 2, 3, 4], i.e.1 x ≈ N X j =1 α j e − c j x ≡ I ( x ) (1.3)which is possible for ℜ ( x ) >
0. We will have to ensure this condition is always met possibly adding a factorwhich changes sign when appropriate.The exponential sum approximation has not been investigated too thoroughly for complex denominators.In fig. 1 The approximation breaks down for small values of ℜ ( x ) as expected, but accuracy is not impactedby an imaginary part as only ℜ ( x ) > ∗ [email protected] /z and its exponential sumapproximation with z = x + iy sing the following complex sign functioncsgn( s ) = − ℜ ( s ) < ℜ ( s ) > ℑ ( s )) ℜ ( s ) = 0 (1.4)and expanding the exponential function as a power series we obtain N R − N P = 12 πi I C ds (cid:2) K ′ ( s ) Z ( s ) + K ( s ) Z ′ ( s ) (cid:3) N X j =1 n X k =0 α j csgn ( f ( s )) ( − k k ! c kj K k ( s ) Z k ( s )csgn k ( f ( s ))= 12 πi N X j =1 n X k =0 I C dsα j csgn k +1 ( f ( s )) ( − k k ! c kj K ′ ( s ) K k ( s ) Z k +1 ( s )+ 12 πi N X j =1 n X k =0 I C dsα j csgn k +1 ( f ( s )) ( − k k ! c kj K k +1 ( s ) Z ′ ( s ) Z k ( s ) = Z π dφ K ( φ ) (1.5)The powers of Z ( s ) can be expressed in terms of z ( t ) using the Mellin convolution theorem Z ( s ) = Z ∞ dtz ( t ) t s − (1.6) Z ( s ) = Z ∞ dt Z ∞ du z ( u ) z (cid:18) tu (cid:19) u − t s − (1.7) Z ( s ) = Z ∞ dt Z ∞ du Z ∞ du z ( u ) z (cid:18) u u (cid:19) z (cid:18) tu (cid:19) u − u − t s − (1.8) Z k ( s ) = Z ∞ dt Z ∞ du . . . Z ∞ du k − t s − z ( u ) z (cid:18) tu k − (cid:19) u − k − Y j =1 z (cid:18) u j +1 u j (cid:19) u − j +1 (1.9)In appendix B a short Maple program is given which can be used to test the formulas given above. Similarly,exploiting standard rules for the Mellin transform Z ′ ( s ) = Z ∞ dt ln( t ) z ( t ) t s − (1.10) Z ′ ( s ) Z ( s ) = Z ∞ dt Z ∞ du ln( u ) z ( u ) z (cid:18) tu (cid:19) u − t s − (1.11)(1.12)If the behavior of the csgn-function is non-trivial for the function to be investigated it may be approxi-mated continuously by csgn( x ) ≈ tanh( x/ǫ ) (1.13)with precision increasing as ǫ −→ x ) = − iπ Z ∞ t ixπ − t − dt (1.14)The integral converges if − π/ < ℑ ( x ) < We use the zeta function in the form [5, 6] ζ ( s ) = 2 s − (1 − − s ) Γ( s + 1) Z ∞ t s cosh ( t ) dt (2.1)which converges for ℜ ( s ) > −
1. The zeta function has been investigated using the argument principle beforeby many authors [7]. The representation in eqn. 2.1 is written in the form of eqn. 1.2. Since the factor K ( s ) in front of the integral has no roots or poles by itself with the exception of the known pole at s = 1 itwould be sufficient to set K ( s ) = 1. However, for the present purpose the full factor is retained in order tostay in a numerically favorable range achieving sufficient accuracy in the exponential sum approximation. φ/ (2 π ) 12 πi dzdφ · f ′ ( z ) f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z = z Reiφ dzdφ · f ′ ( z ) I z )2 πi (cid:12)(cid:12)(cid:12)(cid:12) z = z Reiφ dzdφ · f ′ ( z ) I z )2 πi (cid:12)(cid:12)(cid:12)(cid:12) z = z Reiφ K ( φ )0 0 . . i . . i . . i . . i / . . i − . . i − . . i − . . i / − . . i − . . i − . . i − . . i / − . . i − . . i − . . i − . . i / − . − . i − . − . i − . − . i − . − . i / − . − . i − . − . i − . − . i − . − . i / . − . i . − . i . − . i . − . i / . − . i . . i . . i . . i / . . i . . i . . i . . i Table 1: R = 0 . z = 0 .
57 + 1 . i In table 1 we contrast the integrand in the argument principle with various steps towards the finalapproximation. Integration is performed on a circle with radius R = 0 . z = 0 .
57 + 1 . i notenclosing any roots. The second column is the integrand and factor of eqn. 1.1 with the final dφ -integrationmissing. The coefficients used in the exponential sum approximation can be found in table 2 in the appendix.In the third column 1 /f ( z ) is approximated by I ( z ) which contains the exponential sum approximation.This is approximated further by I ( z ) where the exponential function has been expanded in a power seriesup to linear order. Finally, in the fifth column the powers of Z are expressed by Mellin convolutions givenby eqn. 1.9 and 1.12. Mathematica code producing the results in table 1 can be found in appendix C.Looking at the error introduced in each step we find that the expression by Mellin convolutions (cf.column 4 and 5 in table 1) works fairly well. The largest error is introduced by the exponential sumapproximation which could be reduced by using more exponential terms (higher value of N in eqn. 1.3).Ultimately, for arbitrarily high precision the number of terms in the expansion of the exponential functionneeds to be increased as well, though (higher value of n ). Each new term introduces integrals of one moredimension which makes them increasingly hard to evaluate numerically. We presented a method which evaluates the number of roots minus the number of poles enclosed in a regionusing the argument principle focusing on function which can be expressed as Mellin transforms of simplefunctions. The method was devised to work with the latter (simpler) function which was made possible bymaking use of the exponential sum approximation and the expansion of the exponential function in a powerseries. The powers could be expressed in terms of Mellin convolutions of the simpler function. Because ofthe high dimension of the involved integrals the method may not be feasible for high precision. However,since depending on the function of interest the integrands may be simple it may be possible to come upwith analytical estimates which may or may not exclude roots in a given region. Acknowledgments
We acknowledge support by Wolfram Research having provided assistance with Mathematica and freemaintenance thereof.
References [1] Wolfgang Hackbusch. Computation of best l ∞ exponential sums for 1 /x by remez’algorithm. Computingand Visualization in Science , , 1–11 (2019).[2] William McLean. Exponential sum approximations for t − β , 2016, 1606.00123.[3] William McLean. Exponential sum approximations for t − β . Contemporary Computational Mathematics- A Celebration of the 80th Birthday of Ian Sloan , page 911–930.[4] Gregory Beylkin and Lucas Monz´on. Approximation by exponential sums revisited.
Applied and Com-putational Harmonic Analysis , , 131 – 149 (2010). Special Issue on Continuous Wavelet Transformin Memory of Jean Morlet, Part I.[5] Michael S. Milgram. Integral and series representations of riemann’s zeta function, dirichelet’s etafunction and a medley of related results, 2012, 1208.3429.[6] Michael S. Milgram. Integral and series representations of riemann’s zeta function and dirichlet’s etafunction and a medley of related results. Journal of Mathematics , , 1–17 (2013).[7] Tomas Johnson and Warwick Tucker. Enclosing all zeros of an analytic function — a rigorous approach. Journal of Computational and Applied Mathematics , , 418 – 423 (2009). A Coefficients for the exponential sum approximation i α i c i .
048 0 . .
235 0 . .
852 0 . .
737 2 . B Maple Code Testing Eqn. 1.8 The following Maple code computes the third power of the integral in eqn. 2.1 for s = 0 . . . s = 0 . − . i we obtain 0 . . i and 0 . . i , respectively.z := unapply ( t / cosh ( t ) ˆ 2 , t ) ;i n t e g r a n d := unapply ( z ( u2/u1 ) ∗ z ( t /u2 ) ∗ z ( u1 )/ u1/u2 , u1 , u2 , t ) ;expr1 := I n t ( i n t e g r a n d ( u1 , u2 , t ) , u1 = 0 . . i n f i n i t y ) ;expr2 := I n t ( expr1 , u2 = 0 . . i n f i n i t y ) ;I n t ( expr2 ∗ t ˆ( s −
1) , t = 0 . . i n f i n i t y ) ;su b s ( s =0.4 ,%);e v a l f (%);Z a l t := unapply ( Zeta ( s ) ∗ (1 − − s ) ) ∗ GAMMA( s +1)/2ˆ( s −
1) , s ) ;( Z a l t ( 0 . 4 ) ) ˆ 3 ; Code tested using Maple 2019.2 for Mac OS X Mathematica Code Producing Table 1 z [ t ] := t /Cosh [ t ] ˆ 2K[ s ] := 2ˆ( s − − − s ) ) /Gamma [ s +1]Kp [ s ] := Ev alu ate [D[K[ s ] , s ] ]f [ s ] := Zeta [ s ]Csgn [ x ] := S ign [ Re [ x ] ]Zetap [ s ] := Ev alu ate [D[ Zeta [ s ] , s ] ]alp h a = { } alp h a = { } c = { } c = { } z0 =57/100+157/100 ∗ IR=1/10i n f = \ [ I n f i n i t y ]n j=4n=1Z1 [ s ] : = N I n t e g r a t e [ z [ t ] ∗ t ˆ( s −
1) , { t , 0 , i n f } , Work in gPrecision − >
50, AccuracyGoal − > ∗ z [ u1 ] / u1 ∗ t ˆ( s −
1) , { u1 , 0 , i n f } , { t , 0 , i n f } ,Work in gPrecision − >
50, AccuracyGoal − > ∗ z [ t ] ∗ t ˆ( s −
1) , { t , 0 , i n f } , Work in gPrecision − > − > ∗ z [ t /u1 ] ∗ z [ u1 ] / u1 ∗ t ˆ( s −
1) , { u1 , 0 , i n f } , { t , 0 , i n f } ,Work in gPrecision − >