TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT
.. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the zeros of all partial sums of the expasio (-z) + a z + a z +... + a z +... a...( ).4... 8 5 6 5 8 ()! (!)... all lie outside the uit circle z. It is easily verified that this is ideed true for the first few partial sums but I have bee uable either to devise a fuctio-theory proof for the geeral case or to fid i the literature ay geeral theorems o the zeros of partial sums which could be applied to this expasio. The proof to be preseted here proceeds i the followig way. We show that the real part of ay partial sum is positive o the uit circle. Beig harmoic it is therefore also positive iside the circle ad hece the partial sum caot vaish there. Now the real part o the circle is a fuctio of the polar agle ad we shall show separately ad by etirely differet methods that it is positive (a) for a rage ad (b) for a rage π where. that covers the whole rage < π. No sigle method has bee foud the trigoometric sum ivolved by a related itegral. make use of trigoometric maipulatios. For (a) we approximate to For (b) we There is othig fixed about ad : may variatios of (a) ad (b) ca be exhibited leadig to differet values of ad. All that is ecessary is that we obtai values such that < thus makig the two rages of overlap. () ()
It is ot difficult to covice oeself that the real part of every partial sum is positive o the uit circle: if for example oe tabulates the values of + a cos +... + a cos. for betwee ad 8 at itervals of say ad for... up to say or the result stated Is see to be highly likely. But closer examiatio of the values suggests that a eve stroger result holds amely that these values are greater tha or equal to ½ with equality oly for ad 8. It turs out that it is almost as easy to prove this by methods (a) ad (b) as to prove the weaker result ad we shall therefore proceed to do this. Thus lettig s () + a cos + a cos +... + a cos. () we shall show that s () (4) It is clearly oly ecessary to cosider the rage π.. Rage. The sum s ( ) may be writte for ay fixed s () ' O ak cos k Σ f(x k ) (5) where x k k ad f(x) a ( x / ) c o s x a(t) beig a cotiuous fuctio takig the value a k whe t k that the first term is halved.) (The prime o Σ deotes It is clear therefore that s () may be roughly regarded as a trapezoidal approximatio to the itegral f(x) dx take betwee appropriate limits ad the value of the itegral
gives a estimate for s (). Obviously for the estimate to be useful should ot be too large. The formula () for a is too complicated to be practicable here so we obtai a approximatio usig Stirlig's formula. For our purpose we eed both a upper ad a lower estimate for a ad these are give by the iequalities. ( ) a. 8 < < ( π ) ( π ) (6) These are proved as follows. By oe form of Stirlig's formula The () gives λ! ( ) (π) exp( ) < λ' < e 6 ) 4 λ 88 ()! ( (4π) exp( ) < λ <. λ a exp ( + ) < λ <. (7) (ππ 8 8 6 Now usig Tylor s Theorem with Lagrage remaider exp ( µ / 8 ) > exp ( ) + < µ < 8 8 8 > 7 / 8 8 + 8 while exp( λ / ) > + λ > 6. 8 8 8 It follows that exp( ) exp( ). 5 8 + λ > 8 8 > 8 givig the first iequality i ( 6 ). The secod follows at oce from (7). We ow let f(x) (6) the terms of () satisfy where cos x ad x k k with fixed. The by x a k cos k r k ( ) f(x k) k (8) π r k 5 if f(x 6 k ) > if f(x k ) <.
the If we write p [(p-)π/)] p... cosk k [ ][ + ][ 4 +l 5 ]... (9) cosk k [l + ][ + 4 ]... () With the help of (8) we shall prove that there exist µ ad σ with µ σ such that for ( ) s cosk o () O 'ak > μ > ad q s ( s ( a cosk > σ q... () q + k The clearly ( ) > q... ad it will follow at oce that For if while if S q s( )> () [ q + q+ ] s ( ) s q ( ) by (9) [ q- + q ] s ( ) s q ( ) by (). To obtai a iequality of the form () let us write for s q simplicity S ( ) - s ( ) Q q P. The Ssi½ - a p+ si (P + ) + (a P+ -a P+ )si( (P + ) ) +... + (a Q- -a Q )si( (Q ) Now + a Q si( (Q + ). 4. Thus ad hece P π δ where δ <. (4) -si (P + ) cos ( δ ) cos Ssi > a p+ cos - (a p+ -a p+ ) -... - (a Q- -a Q ) - a Q
i.e. S > ap ta +. 4 Now by (6) ad (4) a ( P + < ). It follows that π S > ( ) ta σ( ) say π 6 4 ad sice (8) is a icreasig fuctio s > - σ ( ) if (6) which is () if we take a σ( ). For () we must use a etirely differet procedure. observe first that f"(x) x -/ (six- (x-/4x) cos x) which has just oe root i ( Π/) amely at x γ π.76 ad is positive to the left of this poit. Thus the graph of f(x) is cocave upwards i (y) ad ay trapezoidal sum of values of f(x) i (y) exceeds the correspodig itegral. By (8) we deduce a similar iequality for a sum of terms a k cosk where k < γ. We We caot however use cocavity i this way i ( γ π/) where f(x) is cocave dowwards. Istead we use the stadard result a+ h h (f(a) + f(a + h)) f(x)dx < h (max f"(x) i (a a + h)) a for the error i trapezoidal quadrature- [ γ / ]. Now let The it is easy to see from the above cosideratios that 5 (5) s ( ) > + cos + ( ( π) π/ π/ f(x)dx 6 f(x)dx) + where + ( 5 A π 6 B C) A ( f(x ) f(x)dx) π/ B ( f(x ) f(x)dx ( 7)
6 c (( - ) + δ ) M M max f " ( x ) i ( π/). We deal with A B ad C i tur ad shall assume that The < < π/4. A 7 cos x six dx is a icreasig fuctio of for its derivative is si + Thus A - 7 x sixdx> π π Next writig δ δ' < < 4 si δ' B < ( ) π ( ') δ π / < ( ( + ) (π - δ' ) π < ( π / ) π < 4( π δ' ). π / δ ' ( ( 6 5π ) π + ) >. cos x dx ) ( ) π ) π x / dx π > 7π 4 6 Agai δ π - < > γ - > γ ad M max f"(x) i ( 7π π ). Now if we write 6 v(x) x / f"(x) - six- (x ) cos x it is easily see that v ' ( x ) < i ( 7π π ). It follows that Further we either have or < 6 v( 7 ) v( π 6 4 x M < 7 / ( π ) max π ) <.559. γ 6 < - < π ad δ <- π - γ <.77 γ ad δ <.
Thus if < π/4 c < π ( π +. 77 ) M <. 556 π. Fially collectig results ad usig the values ([]. p.9) Where π π π / π / f (x) dx f (x) dx C(x) C( ) x C( ) cos ( π t < > ) dt. 4545. 94 we obtai from (7) s ( ) > µ( ). 85 + cos - '87 < < π 4 ad sice µ( ) is a decreasig fuctio s ( ) > µ( ) if o < < < π. (8) 4 Comparig (6) ad (8) with () ad ( ) the desired result () will ow follow if we ca fid < π/4 such that µ( ) σ( ). A suitable value (ad oe ear the maximum possible) is.69 for µ(.69) >. while σ(.69) <.9.. Rage l π. By multiplyig () by si it readily follows that if 7. where with a si ( + ) s ( ) φ ( ) + si b k si ( + ) ( ) φ + 4 si b a a a k a k k k k k k. ( 9 ) ( ) ( ) Thus the b's like the a's form a positive decreasig sequece. They are i fact the coefficiets i the expasio (-z) (-Z)(-Z) We eed also to rewrite (9) as - b z - b z -.... s ( ) Φ p ( ) + ψ p ( ) p< ()
Where The by (). ψ p ψ ( ) p p+ ( ) b k si(k ) + a si( + ) ( ) si bk + a p+ < si Notig also (9) it follows that a p s () Φ p () - si ap ( 4) si p (5) 8. with equality oly if is odd p ad π. Let x deote cos. The i the rage - x s ( ) x + with equality oly whe x - (6) 4 8 4 4 s () x + x + (x + ) + >. (7) Now take p i (5). Sice for ay k si( k ) U (x ) si k + where U is the Chebyshev m si si U k (x ) polyomial x + si si of 5 the sec 4x Thus the right had side of (5) becomes y/ where We may write this as ad y 4x + 6x + 9-5. ( x) y (x + ) + u u x + 8 - y (x - ) + w w x + 8 - od + x 5 ( x) kid + 5 ( x). we ad sice u"< ad w"< i (-) the graphs of u ad w are both have cocave dowwards ad therefore lie above ay chord. Now U(-) u() 8-5 > whece u > i [- ]. Similarly w() > w( 8) 6-5 > so w > i [ 8] It follows that y> i [- 8] ad sice cos( 644) < 8 we have by (5) (6) ad (7)
9. s ( ) o.644 π (8) with equality oly whe ad π. Sice we have proved () with.69> ( 4 ) follows. Remark. It is i fact the case that y> i a wider iterval tha (-.8) amely i (- ξ) where ξ.88... this beig the oly root of y i (-). This is ot difficult to prove with the help of Rolle's theorem ad gives a lower value of but the above method is simpler ad gives a low eough for our purpose. The same applies to several other possible variatios o the use of (5). Remark. Suppose it has bee proved that for some value of p a p φ p ( ) α. si (9) The by (5) it follows that s ( ) α p. () si( k Now ) k decreases steadily from the value k - at si to zero at π/(k-). Thus if we take m [ + ] α π the α π/(m-l) ad φ m () >φ m (α) <α. Now suppose that m > p. (This is ot ecessarily the case for if a is arbitrarily icreased towards π m decreases towards. If however a has the smallest value for which (9) holds i.e. is such that φ p (α) a p / si α the probably m > p follows but I have bee uable to prove this.) The φ m (α) > φ p (α) sice all terms of φ m (α) are positive ad they iclude the terms of φ p (α) while a m < a p. () Thus if we defie α' by a m si α' () φm ( α )
. we have if α ' < si α α ' we a p < si φ ( α ) have φ m ( ) > α by a si m ( 9 ) α ' i.e. α ' a si m < α. Moreover so that () ca be improved (as regards ) to s () > α m. Further if < m ad k + (k- )α π ad it follows by (9) that s () > α < m ad thus fially s () α' p () i.e. () with a replaced by α' < α. The whole process ca ow be iterated: puttig α' i place of a i () gives m' certaily m ad puttig α' ad m' i () i place of a ad m gives α" certaily < α' (sice φ m (α') φ m (α') > φ m (α)). I this way we produce a strictly decreasig sequece a α' α".... If this sequece could be proved to coverge to zero we would have a proof that s () V p. I fact however takig p or the sequece does ot appear to coverge to zero so we caot use this method as a meas of avoidig the use of the method of sectio. This covergece behaviour is associated with the slow rate of covergece to zero of the sequece a ad ideed this is at the root of failure of several other methods that have bee tried for solvig the preset problem. Remark. (4) ad hece (5) depeds o the b k s ad a beig all positive i.e. the a k 's formig a positive decreasig sequece ad the same method could be used to prove s () for r wheever the coefficiets have this property. Similarly (5) also holds i all such cases ad the method of sectio ca be attempted wheever bouds o a as i (6) are kow.
. 4. Coclusio. We have proved that the trigoometric series () has the simply property ( 4 ) whe the a are the coefficiets i the Maclauri series ( ) ad hece as a corollary that all partial sums of () have all their zeros outside the uit circle. Ispite of its complicated ature the preset proof is put forward i the hope of stimulatig the search for a better oe for surely such exists! REFERENCES. M. ABRAMOVITZ ad I.A. STEGUN "Hadbook of mathematical fuctios". Applied Math. Ser. 55 U.S. Govt. Pritig Office Washigto D.C. 965.. A. TALBOT The uiform approximatio of polyomials by polyomials of lower degree. Tech. Report TR/44 Dept of Mathematics Bruel Uiversity 974.