ON TURAN'S INEQUALITY FOR ULTRA- SPHERICAL POLYNOMIALS
|
|
- Griffin Phillips
- 5 years ago
- Views:
Transcription
1 ON TURAN'S INEQUALITY FOR ULTRA- SPHERICAL POLYNOMIALS K. VENKATACHALIENGAR and s. K. LAKSHMANA RAO 0. Introduction. Some years ago P. Turan noticed the interesting inequality (0.1) An(X) = [Pn(x)]2 - Pn+l(x)Pn-l(x) 0,» 1, X \ g 1, where P (x) is the Legendre polynomial of order n and the above inequality has been extended in recent years to the case of some other orthogonal polynomials as well as the ordinary and modified Bessel functions [l; 2]. B. S. Madhava Rao and V. R. Thiruvenkatachar [3] deepened the inequality (0.1) by showing that d2 2 T d I2 (0.2) A (x) = - - Pnix) Z 0. 2 n(n + 1) L J The inequality (0.1) has also been generalized to the case of ultraspherical polynomials P M(x) in the form (0.3) A,.x(«) = [FnAx)]2 - Fn+i.x(x)Fn-i,x(x) ^ or < 0, according as \x} ^ or >1, where n,x(x) =PnMix)/Pnx\l). V. R. Thiruvenkatachar and T. S. Nanjundiah [4] and A. E. Danese [5] have obtained series of positive functions for A (x) and its analogue in the case of the ultraspherical, Laguerre and Hermite polynomials. In the present paper we deepen the above results on ultraspherical polynomials by determining the signs of the first and second derivatives of AnX)ix)=[PnX)ix)]2-Pn+1ix)Pnx!.lix) for various values of X and x. We notice first of all that danx)ix)/ can be represented as a numerical multiple of the Wronskian Pn+iix)dPnxllix)/ Pn-i(x)idPn+lix)/) and express it also via the Christoffel- Darboux formula as a series of functions of constant sign. The sign of da!nx)ix)/ for various values of X and x follows easily from this. We determine the sign of (X l)xida )ix)/) for a general value of n and X and show that id2/2)anx)ix) ^0 according as X>1 or 1/2 ^X<1, thus extending the result of B. S. Madhava Rao and V. R. Thiruvenkatachar to the case of ultraspherical polynomials. For integer values of n we obtain a series of positive functions for id2/2)anx\x). Next we show that (1-x2)AlM(x) decreases steadily in (0, 1) when KX^3/2. From the signs of da^ix)/, Received by the editors August 9, 1956 and, in revised form, April 1,
2 1076 K. VENKATACHALIENGAR AND S. K. LAKSHMANA RAO [December d2anx)(x)/2 we prove the concavity of A,x(x) for 0 <X^ 1/2 and also arrive at inequalities for An,\(x) in the range \x\ : 1, which are not only simpler but also cover a wider range of X than the corresponding inequalities given by O. Szasz [2]. Finally we determine an integral representation for An(x). 1. Expressing da^\x)/ as a Wronskian. We append below the well known relations satisfied by the ultraspherical polynomials [6] for frequent use in the following. d2y dy m (1.1) (1 - x2) - (2X 4- l)x + 11(11 + 2X)y = 0, y = P (x) 2 (1.2) npn\x) - 2(m + X - l)xpnx-i(x) + (n + 2X - 2)Pn-i(x) = 0, w = 2 (1.3) -P»' (*) = /(r(x + r)/r(x))p( "!;) (*), r ^ n r ex) d (\) o (X) (1.4) npn (x) = x Pn (x)-pn-i(x), (X) d (x) d a) (1.5) (n + 2X)Pn (X) = Pn+l(x) - X Pn (x), (xi d <x) d (x> (1.6) 2(n + X)P (x) = Pn+i(x)-P -i(x), (1.7) (1 - x2) Pn\x) = - nxpt(x) + (n+2\- l)p»_i(*), (1.8) (1 - x2) Pn\x) = (» + 2X)*P,<iX>(.T) - («+ l)p (+i(*). Differentiation of (1.9) A? \x) = [PlX,(x)]2- P%i(x)P?2i(x) yields (1.10) An\x) = af\x) - Bn\x) where M(.) = P(:\x) ± P?(X) - PZ(X) y Pn+\(X)
3 i957l ULTRASPHERICAL POLYNOMIALS 1077 and We have then»)., rx)., d (X) (X) d <x> 8n (X) = Pn+l(x) Pn-l(x) - Pn (x) Pn (x). Pn+i(x) Pn-i(x) (1.11) an (x)+8n (x) = d (x) d (X). Pn+l(x) Pn-l(x) Also nan (x) (n + 2X)/3 (x) = 2(n + \)PnX\x) Pn\x) - npnli(x) P i(x) (X) d (x) - (n + 2\)Pn+i(x) P _i(*) = 2(«+ X)P^X)(^) - Pn\x) - (n +2\- l)p i(x) - Pn+\(X). (X), d (x>, s - (n + l)pn+i(x) Pn-i(x) - (2X - 1) \Pn+\(x) Pn-l(x) ~ Pnll(X) P +\(x)\. { ) The first three terms together are found to vanish on using (1.4), (1.5), (1.6) and we get nan\x) - (n + 2\)8n\x) = - (2X - 1)[«?'(*) + Bn\x)\, so that there follows the relation (1.12) (n + 2X - l)at(x) = (n+ l)^x>(x). Solving (1.11) and (1.12) for anx)(x), 8 )(x) and using (1.10), we have Pn+l(x) Pn~l(x) d (X) 1 X (1.13) An (x) = - d (x) d a,. n + X Pn+i(x) P _i(at) ax ax
4 1078 K. VENKA'I ACHALIENGAR AND S. K. LAKSHMANA RAO [December From (1.3) and (1.13) we can also express the first derivative (X) r dr (M ~12 dr (>,) dr (xi Ankx) m - Pn \X) - PUl(x) Pnll(x) l_r J r r of as a constant times the Wronskian of P _+r+i(x) and P^X-iix). 2. Series of functions of constant sign for danx\x)/. Using (1.2) in succession we can write (2.1) Pn+\(X) = anxpnll(x) + KP^x) + CnP^x), where (2.2) an = 4(«+ X)(w + X- 1) -' nvn + 1) («+ X)(»+2X-2)(» + 2\-3) nin + 1)(» + X - 2) Hence the determinant t/ ^ Ux, y) = P +\(x) ex) P^ix) (x) Pn+iiy) Pn-iiy) can be written in the form anx Pn-lix) + bnpn-lix) + CnPn-iix) Pn-\(x) anypnliiy) + bnpn-i(y) + cnpn-ziy) P i(y) and so we get.2 2 (X) (X) = an(x y )Pn-i(x)Pn-i(y) cnon-i(x, y) 2 2 (X) (X) (2.3) Bnix, y) + cjn-i(x, y) = o (x - y )Pn_1(x)Pn_1(y). On observing that we obtain Kix^ti PB+i(*) Pn-i(x) v-*x y - x Pn+iix) P»_i(x) from (2.3), the relation (n + X) d (X) n-2 + \ d a) m -A (x) + cn- A _2(x) = - 2aBx[PB_i(x)J2, 1 X 1 X
5 19571 ULTRASPHERICAL POLYNOMIALS 1079 which can be rewritten in the form m^ ''.Wn (»+2X-2)(«+ 2X-3) d (X) (2.4) A (x)- - An_2(x) n(n + 1) Solving this difference equation d (X), N 8(X- l)r(n+2x- 1) A (x) = -x (n + 1)! 8(n + X - 1) r Cx) l2 = - \ ^n (1 - \)x[pn2i(x)]\ n(n + 1) for da(n)(x)/ we obtain [(-+0/21 (n-2k+ 1)!(» - 2k + X 4-1) w, ' ^ -w-,lxii± n- l^»-»+i(*)j I i_i r(» 2k + 2X + 1) which is the desired series expansion. Differentiation of (2.5) gives a series for (d2/2)anx\x) which has however no particularly elegant form. In the case of Legendre polynomials (X = l/2), (2.5) becomes & 2x K»+l>/2] An(x) = Y (2n - U + 3)[Pn-2k+i(x)]2. n(n + 1) t»i Differentiating d2 this we see that 2 I(n+l)/2] =- E (2» - Ik + 3)P _24+i(x)[P -tt+i(*) + 2xP^2*+,(x)] n(n + 1) t-i which is the same as identity (0.2). From (2.5) it follows that (2.6) sgn a1x>(x)1 = sgn (X- 1)* (X > 0) which amounts to the following property of Anx'(x). The Turdn expression AjX)(x) is an increasing (a decreasing) function for positive (negative) values of x when X > 1. This is to be reversed when 0<X<1. From this we see that when X > 1, A (x) S AB (0) > 0 for all values of x; when 1/2 ^ X < 1, a1x>(x) S A? (l) SO for x g 1; when 0 < X g 1/2, C\x) ^ A^\l) ^ 0 for * gj 1.
6 1080 K. VENKATACHALIENGAR AND S. K. LAKSHMANA RAO [December 3. Sign of (X l)xda^ (x) / for a general index n. For a general index n (which will be restricted in a way later on) we take P \x) as a solution of the differential equation (1.1) which remains regular at x= +1 (or at x= 1) either of which is a regular singular point of the differential equation. We stipulate that the value of the function as well as its slope at x= +1 are positive. Defining Aj,X)(x) as in (1.9) we can again derive the relation (1.13) as before. Multiplying this last relation by (1 x2)xh/2 and differentiating the result, we get ^-\a - xtm t a:\x)] L J 1 _ X Pn+l(x) Pn-l(x) = ^r~x~ 1[(1_,Y^P^m] ±\(l-xt1,2yp^(x)v L J \_ J Simplifying the second row of the determinant by means of the differential equation we are led to d Y 2 X+l/2 d (X) "I (1 - x) An (x) (3.1) l J = 4(l-X)(l-xY-1/2P X+>1(x)PBX_)1(x). Hence the extrema of (1 x2)x+ll2danx)(x)/ occur only at the zeros of Pn+i(x), Pnxii(x). The solution P X)(x) as defined above cannot have zeros on the real segment (1, oo). It increases from a positive value at x= 1. If it attains a maximum value at a point Xi (> 1), we have the conditions (dy/)xi = 0, (d2y/2)zl<q and from the differential equation we notice a contradiction if ra(w + 2X) is positive. Hence P X)(x) is increasing in x> 1 and consequently does not vanish for any x on (1, oc). Let us denote by (a): C*l, 6*2, «3, 03): 0,, 0,, ft, the zeros in ( 1, +1) of P +i(x) and Pfii(x) respectively. From (1.13) we have sgn An (x) \ = sgn PB_i(x) PB+i(x) LdX JI=a LW + X dx Jx=a From (1.7) we may write (I -a2) T^- Pn+\(x) 1 = (n + 2X) PnX\a) LdX J x=a
7 i957l ULTRASPHERICAL POLYNOMIALS 1081 and hence T d (X,, 1 T(X - 1)(» + 2\)PHX!i(a)Pn)(a)l sgn A (x) = sgn --. Lax Jx=a L (w + X)(l - a2) J Using the relation 2(n+\)aPnX)(a) = (n + 2\-l)Plnx±1(a) which follows from (1.2), we get d (X) r n sgn A (x) = sgn [(X - l)(n 4-2\)(n + 2X - l)/a]. In a similar way we observe that sgn [ Af'(x)l Lax JI=,(j = sgn [(X - l)nft/(» +!)] Combining the above two facts we see that whenever n S1 and X>0 [d >\) "I x A (x) = sgn (X - 1). J z=a,s Thus at all possible extrema of (1-x2)x+1/2(a*A^)(x)/a,x), Q^ l)x(danx)(x)/) is positive and hence (3.2) sgn (X - l)x An \x) = + 1, for n S 1, X > 0. For positive integer values of n this is the same as (2.6). 4. Sign of (d2/ox2)a;,x)(x). Differentiating (1.13) which also holds for general n we get D(X) I \ D(X) / \ Pn+l(x) Pn-l(x) d a), s 1 X -A (x) =- d2 (x) d2 (x) 2 «+ X - Pn+l(x) --P _l(x) 2 2 Replacing the second row elements by means of the equations d1 ex) d2 (xi d ai Pn+i(x) = X Pn (x) + (n + 2X + 1) Pn (*), ax2 2 d2 (x> d2 tx) d c\\ p _;(x) = x p '(*) - («-1) - pv(x) 2 2 (these follow on differentiating (1.5) and (1.4)) and replacing the first row elements by means of (1.7) and (1.8) we get
8 1082 K. VENKATACHAL1ENGAR AND S. K. LAKSHMANA RAO (December d2 (X), 1 - X - A (x) = - 2 n + X (n+2x)x.pnx)(x)-(l-x2) pt(x) «xpnx)(x) + (l-x2) P*\x) n + 1 n + 2X - 1 d2 rx) d (xj d2 (x) d (X) x P\ \x) + (ii+2\+l) Pn \x) x--pn\x)-(n-l) P:\x) 2 2 a form in which the right hand side involves only algebraic combinations of P x>(x) and its first and second derivatives. After simplifying this determinant we arrive at the equation (»+l)(«+2x-l) d2 (x,. Vd (x, "j2 -An (x) = 2 P (x) 2(X - 1) 2 \_ J (4.1) + (2X- l)x *[( PnX)(x)J - Pn(X,(x) ~ Pf (*)] Ii -p?\x)jlp?\x)\. ) (4.2) D?(x) = A^x) = \- PlX)(x)]2 - - pv+i(x) - P» (,), Lax J ax we can easily see that DnX)(x) =4X2Anx_+i')(x) and yd?\x) = 2x{x[( p1x)(x))2 - PIX)(X) ^PlX)(x)] - A)-ip.WW} We may therefore (4.3) write (4.1) in the form (»+l)(»4-2x-l) d2 w -An 2(X - 1) 2 (X) T d rx) "I2 d (x+d = 2 - p; \x) + 2x(2x - i)x - a;j; (*). Lax J Either of (4.1) and (4.3) is an extension of (0.2) to the case of ultraspherical polynomials. When n is a positive integer we may replace the last term on the right side of (4.3) by means of the series (2.5)
9 i957l ULTRASPHERICAL POLYNOMIALS 1083 and so arrive at the following series for d2abx>(x)/2 (»+l)(»+2x- 1) d2 (x,/ N - -A (x) 2(X - 1) 2 r d (x) "I r(«-i- 2X) = 2 PB \x) \2 + (2X - 1) - 16X2x2 \_ J nl "» 1 in - 2k) l{n - 2k + X + 1) u+1) 2^ -:-: LPB-2fc(x)J-. [ Tin - 2k + 2X + 2) From (3.2) it follows that \xdan\\l)ix)/>0 Using this in (4.3) we deduce that (a «d* a(xv ^ /+1 (X>1)' (4.5) sgn-ab (x) = < B 2 l-l (1/2 X < 1). for w^2, X>-l/2. Hence ABX)(x) is a convex function of x for X> 1 and a concave function of x for l/2gx<l. 5. Decreasing nature of (1 x2)a X)(x). From (3.1) we know that A X)(x) satisfies the differential equation r d'2 d i cm (5.1) ^(1 - x2) - (2X + l)x - + 4(1 -X)jAB (*) If (1 x2)anx)(x) is denoted by y, we get = 4(l-X)[P:X)(x)]2. (5.2) = (1 - x ) - An (x) - 2xA (x), -= (1 - x ) -A (x) \x AB ix) 2AB (x) We find from (5.1) and (5.2) that y satisfies the differential equation (5.3) [(1-^ i-(2x-3)^x-(4x-6)]^ = (4X - 6)x2ABM(x) + 4(1 - X)(l - x)[pn\x)]\ Now y(0)=a;x)(0) is positive when X>0 and y(l)=0. When X>1, dy/ is negative for all x > 1 and hence y decreases for x > 1 when X> 1. We shall now see that y steadily decreases even in (0, 1) when KXg3/2. On using (2.6) we have idy/)x=o = 0. From (5.3) we have (d2y/ix2),=o = (4X-6)AiX)(0)-r-4(l-X)[PiX)(0)]2 and so when
10 1084 K. VENKATACHALIENGAR AND S. K. LAKSHMANA RAO [December l<x 3/2, id2y/2)x=o is negative. Hence when KX 3/2, v reaches a maximum at x = 0 with the maximum value =ABX)(0) and decreases in the neighbourhood of x = 0. Also (dy/)x=i= 2A X)(1) is negative when X>l/2 and so y(x) is decreasing in the neighbourhood of x= 1 also when X> 1. If y(x) is not steadily decreasing in (0, 1) it must reach a minimum value at some point x = a<l, then increase to a maximum at some point x=j8 between a and 1 and then decrease again. We show that this possibility cannot occur when 1 <X 3/2. From the conditions for minimum, we have at x = a, (dy/) =0 and (d2y/2)>0 and from the differential equation (5.3) we get 2 / d2y\ i m (1 - a ) ( ~ ) = (4X - 6)yia) + (4X - 6)a AB \a) \ 2/x=a + 4(l-X)(l-a2)[PBX>(a)]\ The left hand side is positive while the right hand side is negative. We conclude that y cannot have a minimum anywhere in (0, 1). Therefore v(x) must steadily decrease in (0, 1) when 1 <X 3/2. It is to be expected that for large values of X, y(x) decreases first to a minimum and then attains a maximum in (0, 1) before decreasing again at and near x = 1. Eliminating (1 x2)d2anx)(x)/tix2 in d2y/2 by means of (5.1) we obtain = (2X - 3)x ABX)(x) + (4X - 6)ABX)(x) + 4(1 - X)[P X)(x)]\ 2 Using (3.2) we notice that sgn d2y/2= 1 when 1 <X 3/2. 6. Inequalities for AB,x(x). Let A.x(x) = [p1x)(x)/pbx)(1)]2- [P(n+\ix)/Pn+\il)].[P iix)/p (l)]. From the following well-known relations (see (4), (5)) 2 (X1 '',A n (6.1) A / \ (1 ~ x)dn AB,x(x) =-= (X) 4X \ <M-1>,. - (1 - x )An_i ix) where ^B,x = n(w + 2X) [PBX)(1)]2, and the conclusions section, we have of the previous (a) A,x(x) is steadily decreasing in (0, 1) for 0 < X 1/2; d2 (b) sgn-a,x(x) = - 1 when 0 < X 1/2, 2
11 1957l ULTRASPHERICAL POLYNOMIALS 1085 showing the concavity property of A,x(x). From (a) we can immediately deduce the well known inequality viz., the Turan expression A,x(x)>0 in x <1 when 0<Xgl/2. From (6.1) and the concluding results of (2) we can prove Turin's inequality in full for X>0. We have seen in (3.2) that (3.2) sgn (X- l)x A X)(x) = + 1 for n S 1, X > 0 L J and in (4.5) that d2 (x) (+1 XI>I1 (4.5) sgn-an (x) = < n S 2. 2 l-l 1/2 S X < 1 From the identity D X) (x) = 4X2A xjt"11) (x) it therefore follows that d rx) (6.2) sgn Xx Dn (x) = +1 (n S 2, X > - 1/2), These show the monotonic increasing-decreasing behaviour and the convexity-concavity property of D X)(x) ior various values of X, and enable us to write the following chain of inequalities: When X > 0, Z? X)(0) ^ >lx>(x) ^ A!X)(0)(1 - x/a) + DnX\a) x/a when ^ Dnk\a), -1/2 < X < 0, D X,(0) ^ DnX\x) S DnX)(0)(l - x/a) + D (o) x/a S DnX\a) 0 ^ x ^ a. Taking a = 1 and multiplying the above inequalities by (1 x2) we get the following inequalities for A,x(x) in the range x ;S1, for w^2. A,x(0)(l - x2) =S A,x(x) ^ ran,x(0)(l - x) + (1 - x2) L 2X + 1_ (6.4) = (1 " x2)/(2\ +1) (X > 0), ^fri = l^ttl + A».x(0)(l - *)}(1 - x2) g An.x(x) 2X + 1 (2X + 1 ) g ABlX(0)(l - x2) (-1/2<X<0).
12 1086 K. VENKATACHALIENGAR AND S. K LAKSHMANA RAO [December These are simpler than the inequalities for A,x(x) in O. Szasz's paper [2] and are at the same time an improvement over the corresponding ones in [4]. 7. Integral representation for A (x). From (3.1) we have (7.1) ±\il - x2)^ L AlX)(x)l + 4(1 - X)(l - *tx'\vix) L ox J = 4(1 - X)(l - x2)x-1/2[p lx)(x)]2. If we denote [(l-x2)*-1'2ank)(x)]/x by/ X)(x), then (1 - x2)x+m~ A«\x) = [x{l _ J)f?\x)] + (2X + l)x2f*\x). Hence from (7.1) we get "»(1 - *2)Ax)l + (2X + l)x2 ~fn\x) + 6x/ X>(x) 2 L J which may also be written in the form,, 2vX-I/2r (X),. -,2 = 4(1-X)(1-X) [Pn \x)], d T 2 2 -x+3/2 d (X) "1 rr,(x)/ \i2 x (1 - x ) / (x) = 4(1 - X)x[Pn (x)j. L For Legendre functions (X = l/2) this becomes (7.2) -f-^(l - *2) [*»(*)/*]) = 2x[Pn(x)]2. \ / Hence x2(l x2)(d[ah(x)/x]/) is an increasing (a decreasing) function for positive (negative) values of x and vanishes at x= +1. From (7.2) we have in succession and ~ (An(x)/X) = 2 Ct[Pnit)]2dt, x2(l x2) J 1 (7.3) A (x) = 2x r U Cl[Pnit)]*dt, J 1 «-(l u-) J 1 the latter giving a positive integral representation for A (x). Using the relation
13 i957) ULTRASPHER1CAL POLYNOMIALS 1087 T (1-X2)[^-Pn(x)]2 d, ^ Lax J., - (l-x2)[p (x)] = - 2x[Pn(x)]2, L n(n + 1) we may also write A (x) in the form f r d f} (l-l2)\- Pn(t)\ C r n \-dt J (7.4) A (x) = x [Pn(t)]2 +-;-\r2dt. J x { n(n + 1) References 1. G. Szego, On an inequality of P. Turin concerning Legendre polynomials. Bull. Amer. Math. Soc. vol. 54 (1948) pp O. Szasz, Inequalities concerning orthogonal polynomials and Bessel functions, Proc. Amer. Math. Soc. vol. 1 (1950) pp B. S. Madhava Rao and V. R. Thiruvenkatachar, On an inequality concerning orthogonal polynomials, Proceedings of the Indian Academy of Sciences, Sect. A vol. 29 (1949) pp V. R. Thiruvenkatachar and T. S. Nanjundiah, Inequalities concerning Bessel functions and orthogonal polynomials, ibid. Sect. A vol. 33 (1951) pp A. E. Danese, Explicit evaluations of Turin expressions, Annali di Matematica Pura ed Applicata, Serie IV vol. 38 (1955) pp G. Szegb, Orthogonal polynomials, (1939) pp Mysore University Indian Institute and of Science
AN INEQUALITY OF TURAN TYPE FOR JACOBI POLYNOMIALS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 32, Number 2, April 1972 AN INEQUALITY OF TURAN TYPE FOR JACOBI POLYNOMIALS GEORGE GASPER Abstract. For Jacobi polynomials P^^Hx), a, ß> 1, let RAX)
More informationBounds on Turán determinants
Bounds on Turán determinants Christian Berg Ryszard Szwarc August 6, 008 Abstract Let µ denote a symmetric probability measure on [ 1, 1] and let (p n ) be the corresponding orthogonal polynomials normalized
More informationPositivity of Turán determinants for orthogonal polynomials
Positivity of Turán determinants for orthogonal polynomials Ryszard Szwarc Abstract The orthogonal polynomials p n satisfy Turán s inequality if p 2 n (x) p n 1 (x)p n+1 (x) 0 for n 1 and for all x in
More informationREAL ZEROS OF A RANDOM SUM OF ORTHOGONAL POLYNOMIALS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 27, Number 1, January 1971 REAL ZEROS OF A RANDOM SUM OF ORTHOGONAL POLYNOMIALS MINAKETAN DAS Abstract. Let c
More information(1.1) maxj/'wi <-7-^ n2.
proceedings of the american mathematical society Volume 83, Number 1, September 1981 DERIVATIVES OF POLYNOMIALS WITH POSriTVE COEFFICIENTS A. K. VARMA Abstract. Let P (x) be an algebraic polynomial of
More informationMarch Algebra 2 Question 1. March Algebra 2 Question 1
March Algebra 2 Question 1 If the statement is always true for the domain, assign that part a 3. If it is sometimes true, assign it a 2. If it is never true, assign it a 1. Your answer for this question
More informationON CO-RECURSIVE ORTHOGONAL POLYNOMIALS
ON CO-RECURSIVE ORTHOGONAL POLYNOMIALS T. S. CHIHARA 1. Introduction. Let Pn(x) be a polynomial of degree n(n = 0,1, ) with leading coefficient unity. Then (Favard [2]) a sufficient (as well as necessary)
More informationQuadrature Formulas for Infinite Integrals
Quadrature Formulas for Infinite Integrals By W. M. Harper 1. Introduction. Since the advent of high-speed computers, "mechanical" quadratures of the type (1) f w(x)f(x)dx^ Hif(aj) Ja 3=1 have become increasingly
More informationnail-.,*.. > (f+!+ 4^Tf) jn^i-.^.i
proceedings of the american mathematical society Volume 75, Number 2, July 1979 SOME INEQUALITIES OF ALGEBRAIC POLYNOMIALS HAVING REAL ZEROS A. K. VARMA Dedicated to Professor A. Zygmund Abstract. Let
More informationLinear DifferentiaL Equation
Linear DifferentiaL Equation Massoud Malek The set F of all complex-valued functions is known to be a vector space of infinite dimension. Solutions to any linear differential equations, form a subspace
More informationSome Integrals Involving Associated Legendre Functions
MATHEMATICS OF COMPUTATION, VOLUME 28, NUMBER 125, JANUARY, 1974 Some Integrals Involving Associated Legendre Functions By S. N. Samaddar Abstract. Calculations of some uncommon integrals involving Legendre
More informationHigher-order ordinary differential equations
Higher-order ordinary differential equations 1 A linear ODE of general order n has the form a n (x) dn y dx n +a n 1(x) dn 1 y dx n 1 + +a 1(x) dy dx +a 0(x)y = f(x). If f(x) = 0 then the equation is called
More informationA Note on Extended Gaussian Quadrature
MATHEMATICS OF COMPUTATION, VOLUME 30, NUMBER 136 OCTOBER 1976, PAGES 812-817 A Note on Extended Gaussian Quadrature Rules By Giovanni Monegato* Abstract. Extended Gaussian quadrature rules of the type
More informationƒ f(x)dx ~ X) ^i,nf(%i,n) -1 *=1 are the zeros of P«(#) and where the num
ZEROS OF THE HERMITE POLYNOMIALS AND WEIGHTS FOR GAUSS' MECHANICAL QUADRATURE FORMULA ROBERT E. GREENWOOD AND J. J. MILLER In the numerical integration of a function ƒ (x) it is very desirable to choose
More informationOR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision
More informationTwo special equations: Bessel s and Legendre s equations. p Fourier-Bessel and Fourier-Legendre series. p
LECTURE 1 Table of Contents Two special equations: Bessel s and Legendre s equations. p. 259-268. Fourier-Bessel and Fourier-Legendre series. p. 453-460. Boundary value problems in other coordinate system.
More informationA DUALITY THEOREM FOR NON-LINEAR PROGRAMMING* PHILIP WOLFE. The RAND Corporation
239 A DUALITY THEOREM FOR N-LINEAR PROGRAMMING* BY PHILIP WOLFE The RAND Corporation Summary. A dual problem is formulated for the mathematical programming problem of minimizing a convex function under
More informationMath 2233 Homework Set 7
Math 33 Homework Set 7 1. Find the general solution to the following differential equations. If initial conditions are specified, also determine the solution satisfying those initial conditions. a y 4
More information5.4 Bessel s Equation. Bessel Functions
SEC 54 Bessel s Equation Bessel Functions J (x) 87 # with y dy>dt, etc, constant A, B, C, D, K, and t 5 HYPERGEOMETRIC ODE At B (t t )(t t ), t t, can be reduced to the hypergeometric equation with independent
More informationThis ODE arises in many physical systems that we shall investigate. + ( + 1)u = 0. (λ + s)x λ + s + ( + 1) a λ. (s + 1)(s + 2) a 0
Legendre equation This ODE arises in many physical systems that we shall investigate We choose We then have Substitution gives ( x 2 ) d 2 u du 2x 2 dx dx + ( + )u u x s a λ x λ a du dx λ a λ (λ + s)x
More informationLegendre s Equation. PHYS Southern Illinois University. October 18, 2016
Legendre s Equation PHYS 500 - Southern Illinois University October 18, 2016 PHYS 500 - Southern Illinois University Legendre s Equation October 18, 2016 1 / 11 Legendre s Equation Recall We are trying
More informationSturm-Liouville Theory
More on Ryan C. Trinity University Partial Differential Equations April 19, 2012 Recall: A Sturm-Liouville (S-L) problem consists of A Sturm-Liouville equation on an interval: (p(x)y ) + (q(x) + λr(x))y
More informationCALCULUS JIA-MING (FRANK) LIOU
CALCULUS JIA-MING (FRANK) LIOU Abstract. Contents. Power Series.. Polynomials and Formal Power Series.2. Radius of Convergence 2.3. Derivative and Antiderivative of Power Series 4.4. Power Series Expansion
More informationGenerating Functions
Semester 1, 2004 Generating functions Another means of organising enumeration. Two examples we have seen already. Example 1. Binomial coefficients. Let X = {1, 2,..., n} c k = # k-element subsets of X
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georgia Tech PHYS 612 Mathematical Methods of Physics I Instructor: Predrag Cvitanović Fall semester 2012 Homework Set #5 due October 2, 2012 == show all your work for maximum credit, == put labels, title,
More information1 Solving Algebraic Equations
Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan 1 Solving Algebraic Equations This section illustrates the processes of solving linear and quadratic equations. The Geometry of Real
More informationSeries Solutions of ODEs. Special Functions
C05.tex 6/4/0 3: 5 Page 65 Chap. 5 Series Solutions of ODEs. Special Functions We continue our studies of ODEs with Legendre s, Bessel s, and the hypergeometric equations. These ODEs have variable coefficients
More informationIf(x) - q.(x) I < f(x) - p.(x) I on E where f(x) - p. ., x) are independent.
VOL. 45, 1959 MATHEMATICS: WALSH AND MOTZKIN 1523 consider the ideal generated by dga duel. i Eua...i.,,xj 0 < m < to Eduq.. AujAdm in the ring of all differential forms. This ideal is the system (G).
More informationMA22S3 Summary Sheet: Ordinary Differential Equations
MA22S3 Summary Sheet: Ordinary Differential Equations December 14, 2017 Kreyszig s textbook is a suitable guide for this part of the module. Contents 1 Terminology 1 2 First order separable 2 2.1 Separable
More informationComputation of Eigenvalues of Singular Sturm-Liouville Systems
Computation of Eigenvalues of Singular Sturm-Liouville Systems By D. 0. Banks and G. J. Kurowski 1. Introduction. In recent papers, P. B. Bailey [2] and M. Godart [5] have used the Prüfer transformation
More informationMATH 2250 Final Exam Solutions
MATH 225 Final Exam Solutions Tuesday, April 29, 28, 6: 8:PM Write your name and ID number at the top of this page. Show all your work. You may refer to one double-sided sheet of notes during the exam
More informationOn Some Estimates of the Remainder in Taylor s Formula
Journal of Mathematical Analysis and Applications 263, 246 263 (2) doi:.6/jmaa.2.7622, available online at http://www.idealibrary.com on On Some Estimates of the Remainder in Taylor s Formula G. A. Anastassiou
More informationMath 240 Calculus III
Calculus III Summer 2015, Session II Monday, August 3, 2015 Agenda 1. 2. Introduction The reduction of technique, which applies to second- linear differential equations, allows us to go beyond equations
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include
PUTNAM TRAINING POLYNOMIALS (Last updated: December 11, 2017) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationA SUBSPACE THEOREM FOR ORDINARY LINEAR DIFFERENTIAL EQUATIONS
J. Austral. Math. Soc. {Series A) 50 (1991), 320-332 A SUBSPACE THEOREM FOR ORDINARY LINEAR DIFFERENTIAL EQUATIONS ALICE ANN MILLER (Received 22 May 1989; revised 16 January 1990) Communicated by J. H.
More informationa x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).
You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and
More informationODE Homework Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation. n(n 1)a n x n 2 = n=0
ODE Homework 6 5.2. Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation y + k 2 x 2 y = 0, k a constant about the the point x 0 = 0. Find the recurrence relation;
More informationLIOUVILLIAN FIRST INTEGRALS OF SECOND ORDER POLYNOMIAL DIFFERENTIAL EQUATIONS Colin Christopher. 1. Introduction
Electronic Journal of Differential Equations, Vol. 1999(1999) No. 49, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login:
More informationMOMENT SEQUENCES AND BACKWARD EXTENSIONS OF SUBNORMAL WEIGHTED SHIFTS
J. Austral. Math. Soc. 73 (2002), 27-36 MOMENT SEQUENCES AND BACKWARD EXTENSIONS OF SUBNORMAL WEIGHTED SHIFTS THOMAS HOOVER, IL BONG JUNG and ALAN LAMBERT (Received 15 February 2000; revised 10 July 2001)
More informationCh 4 Differentiation
Ch 1 Partial fractions Ch 6 Integration Ch 2 Coordinate geometry C4 Ch 5 Vectors Ch 3 The binomial expansion Ch 4 Differentiation Chapter 1 Partial fractions We can add (or take away) two fractions only
More informationNOTE ON A SERIES OF PRODUCTS OF THREE LEGENDRE POLYNOMIALS
i95i] SERIES OF PRODUCTS OF THREE LEGENDRE POLYNOMIALS 19 In view of applications, let us remark that every enumerable set (for example the set of rational or algebraic numbers a, b) of substitutions Sab
More informationPolynomial Solutions of the Laguerre Equation and Other Differential Equations Near a Singular
Polynomial Solutions of the Laguerre Equation and Other Differential Equations Near a Singular Point Abstract Lawrence E. Levine Ray Maleh Department of Mathematical Sciences Stevens Institute of Technology
More informationQualification Exam: Mathematical Methods
Qualification Exam: Mathematical Methods Name:, QEID#41534189: August, 218 Qualification Exam QEID#41534189 2 1 Mathematical Methods I Problem 1. ID:MM-1-2 Solve the differential equation dy + y = sin
More informationElectromagnetism HW 1 math review
Electromagnetism HW math review Problems -5 due Mon 7th Sep, 6- due Mon 4th Sep Exercise. The Levi-Civita symbol, ɛ ijk, also known as the completely antisymmetric rank-3 tensor, has the following properties:
More informationUNIFORM BOUNDS FOR BESSEL FUNCTIONS
Journal of Applied Analysis Vol. 1, No. 1 (006), pp. 83 91 UNIFORM BOUNDS FOR BESSEL FUNCTIONS I. KRASIKOV Received October 8, 001 and, in revised form, July 6, 004 Abstract. For ν > 1/ and x real we shall
More informationFOR COMPACT CONVEX SETS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 2, June 1975 ON A GAME THEORETIC NOTION OF COMPLEXITY FOR COMPACT CONVEX SETS EHUD KALAI AND MEIR SMORODINSKY ABSTRACT. The notion of
More information(308 ) EXAMPLES. 1. FIND the quotient and remainder when. II. 1. Find a root of the equation x* = +J Find a root of the equation x 6 = ^ - 1.
(308 ) EXAMPLES. N 1. FIND the quotient and remainder when is divided by x 4. I. x 5 + 7x* + 3a; 3 + 17a 2 + 10* - 14 2. Expand (a + bx) n in powers of x, and then obtain the first derived function of
More informationON THE AVERAGE NUMBER OF REAL ROOTS OF A RANDOM ALGEBRAIC EQUATION
ON THE AVERAGE NUMBER OF REAL ROOTS OF A RANDOM ALGEBRAIC EQUATION M. KAC 1. Introduction. Consider the algebraic equation (1) Xo + X x x + X 2 x 2 + + In-i^" 1 = 0, where the X's are independent random
More information2 Series Solutions near a Regular Singular Point
McGill University Math 325A: Differential Equations LECTURE 17: SERIES SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS II 1 Introduction Text: Chap. 8 In this lecture we investigate series solutions for the
More informationGENERATING FUNCTIONS FOR THE JACOBI POLYNOMIAL M. E. COHEN
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 57, Number 2, June 1976 GENERATING FUNCTIONS FOR THE JACOBI POLYNOMIAL M. E. COHEN Abstract. Two theorems are proved with the aid of operator and
More informationVON NEUMANN'S INEQUALITY FOR COMMUTING, DIAGONALIZABLE CONTRACTIONS. II B. A. LOTTO AND T. STEGER. (Communicated by Theodore W.
proceedings of the american mathematical society Volume 12, Number 3. March 1994 VON NEUMANN'S INEQUALITY FOR COMMUTING, DIAGONALIZABLE CONTRACTIONS. II B. A. LOTTO AND T. STEGER (Communicated by Theodore
More informationProblem 1 (Equations with the dependent variable missing) By means of the substitutions. v = dy dt, dv
V Problem 1 (Equations with the dependent variable missing) By means of the substitutions v = dy dt, dv dt = d2 y dt 2 solve the following second-order differential equations 1. t 2 d2 y dt + 2tdy 1 =
More informationOn certain formulas of Karlin and
On certain formulas of Karlin and Szegö Bernard Leclerc Département de Mathématiques, Université de Caen, 14032 Caen cedex, France Abstract Some identities due to Karlin and Szegö which provide a relationship
More informationTHE MEAN CONVERGENCE OF ORTHOGONAL SERIES. II
THE MEAN CONVERGENCE OF ORTHOGONAL SERIES. II BY HARRY POLLARD Let w(x) be a weight function on ( 1, 1) and pn(x) the corresponding orthonormal polynomials [l, p. 25]('). Then /. i pm(x)pn(x)w(x)dx = ômn
More informationChapter 4. Series Solutions. 4.1 Introduction to Power Series
Series Solutions Chapter 4 In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generation adds a new story to the old
More informationNOTE ON VOLTERRA AND FREDHOLM PRODUCTS OF SYMMETRIC KERNELS*
I93L] PRODUCTS OF SYMMETRIC KERNELS 891 NOTE ON VOLTERRA AND FREDHOLM PRODUCTS OF SYMMETRIC KERNELS* BY L. M. BLUMENTHAL 1. Introduction.] The purpose of this paper is to establish two theorems concerning
More informationswapneel/207
Partial differential equations Swapneel Mahajan www.math.iitb.ac.in/ swapneel/207 1 1 Power series For a real number x 0 and a sequence (a n ) of real numbers, consider the expression a n (x x 0 ) n =
More informationREMARKS ON INCOMPLETENESS OF {eix-*}, NON- AVERAGING SETS, AND ENTIRE FUNCTIONS1 R. M. REDHEFFER
REMARKS ON INCOMPLETENESS OF {eix-*}, NON- AVERAGING SETS, AND ENTIRE FUNCTIONS1 R. M. REDHEFFER If {X } is a set of real, nonnegative, and unequal numbers, as assumed throughout this paper, then completeness
More informationEigenvectors, Eigenvalues, and Diagonalizat ion
Eigenvectors, Eigenvalues, and Diagonalizat ion Section 8.1, p. 420 2. (a) Axl = Xlxl. (b) AXJ = X2x2. (c) Ax3 = X3x3 16. (a) p(x) = X2 + 1. The eige~lvalues are X1 = i and A2 = -i. Associated eigenvectors
More informationENGI 9420 Lecture Notes 1 - ODEs Page 1.01
ENGI 940 Lecture Notes - ODEs Page.0. Ordinary Differential Equations An equation involving a function of one independent variable and the derivative(s) of that function is an ordinary differential equation
More informationNONLINEAR PERTURBATION OF LINEAR PROGRAMS*
SIAM J. CONTROL AND OPTIMIZATION Vol. 17, No. 6, November 1979 1979 Society for Industrial and Applied Mathematics 0363-,0129/79/1706-0006 $01.00/0 NONLINEAR PERTURBATION OF LINEAR PROGRAMS* O. L. MANGASARIAN"
More informationx 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
More informationON DIVISION ALGEBRAS*
ON DIVISION ALGEBRAS* BY J. H. M. WEDDERBURN 1. The object of this paper is to develop some of the simpler properties of division algebras, that is to say, linear associative algebras in which division
More informationMath Assignment 11
Math 2280 - Assignment 11 Dylan Zwick Fall 2013 Section 8.1-2, 8, 13, 21, 25 Section 8.2-1, 7, 14, 17, 32 Section 8.3-1, 8, 15, 18, 24 1 Section 8.1 - Introduction and Review of Power Series 8.1.2 - Find
More informationPractice Final Exam Solutions
Important Notice: To prepare for the final exam, study past exams and practice exams, and homeworks, quizzes, and worksheets, not just this practice final. A topic not being on the practice final does
More informationRodrigues-type formulae for Hermite and Laguerre polynomials
An. Şt. Univ. Ovidius Constanţa Vol. 16(2), 2008, 109 116 Rodrigues-type formulae for Hermite and Laguerre polynomials Vicenţiu RĂDULESCU Abstract In this paper we give new proofs of some elementary properties
More informationGenerating Functions (Revised Edition)
Math 700 Fall 06 Notes Generating Functions (Revised Edition What is a generating function? An ordinary generating function for a sequence (a n n 0 is the power series A(x = a nx n. The exponential generating
More informationSolutions: Problem Set 3 Math 201B, Winter 2007
Solutions: Problem Set 3 Math 201B, Winter 2007 Problem 1. Prove that an infinite-dimensional Hilbert space is a separable metric space if and only if it has a countable orthonormal basis. Solution. If
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
More informationNotes on Special Functions
Spring 25 1 Notes on Special Functions Francis J. Narcowich Department of Mathematics Texas A&M University College Station, TX 77843-3368 Introduction These notes are for our classes on special functions.
More informationON THE EQUATION Xa=7X+/3 OVER AN ALGEBRAIC DIVISION RING
ON THE EQUATION Xa=7X+/3 OVER AN ALGEBRAIC DIVISION RING R. E. JOHNSON 1. Introduction and notation. The main purpose of this paper is to give necessary and sufficient conditions in order that the equation
More informationSeries Solutions of Linear ODEs
Chapter 2 Series Solutions of Linear ODEs This Chapter is concerned with solutions of linear Ordinary Differential Equations (ODE). We will start by reviewing some basic concepts and solution methods for
More informationPower Series Solutions to the Legendre Equation
Department of Mathematics IIT Guwahati The Legendre equation The equation (1 x 2 )y 2xy + α(α + 1)y = 0, (1) where α is any real constant, is called Legendre s equation. When α Z +, the equation has polynomial
More informationISOMORPHIC POLYNOMIAL RINGS
PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 27, No. 2, February 1971 ISOMORPHIC POLYNOMIAL RINGS D. B. COLEMAN AND E. E. ENOCHS Abstract. A ring is called invariant if whenever B is a ring
More informationAPPENDIX : PARTIAL FRACTIONS
APPENDIX : PARTIAL FRACTIONS Appendix : Partial Fractions Given the expression x 2 and asked to find its integral, x + you can use work from Section. to give x 2 =ln( x 2) ln( x + )+c x + = ln k x 2 x+
More informationLecture Notes for Chapter 9
Lecture Notes for Chapter 9 Kevin Wainwright April 26, 2014 1 Optimization of One Variable 1.1 Critical Points A critical point occurs whenever the firest derivative of a function is equal to zero. ie.
More informationFirst-Order ODE: Separable Equations, Exact Equations and Integrating Factor
First-Order ODE: Separable Equations, Exact Equations and Integrating Factor Department of Mathematics IIT Guwahati REMARK: In the last theorem of the previous lecture, you can change the open interval
More information(U 2 ayxtyj < (2 ajjxjxj) (2 a^jj),
PROCEEDINGS Ol THE AMERICAN MATHEMATICAL SOCIETY Volume 59, Number 1. August 1976 THE MEANING OF THE CAUCHY-SCHWARZ- BUNIAKOVSKY INEQUALITY eduardo h. zarantonello1 Abstract. It is proved that a mapping
More informationand verify that it satisfies the differential equation:
MOTIVATION: Chapter One: Basic and Review Why study differential equations? Suppose we know how a certain quantity changes with time (for example, the temperature of coffee in a cup, the number of people
More information(1) L(y) m / - E/*(*)/ = R(*),
THE APPROXIMATE SOLUTION OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS1 R. G. HUFFSTUTLER AND F. MAX STEIN 1. Introduction. We consider the best approxiation by polynoials P (x) of the solution on [0, l],
More informationProblem Set 5: Solutions Math 201A: Fall 2016
Problem Set 5: s Math 21A: Fall 216 Problem 1. Define f : [1, ) [1, ) by f(x) = x + 1/x. Show that f(x) f(y) < x y for all x, y [1, ) with x y, but f has no fixed point. Why doesn t this example contradict
More information("-1/' .. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS. R. T. Rockafellar. MICHIGAN MATHEMATICAL vol. 16 (1969) pp.
I l ("-1/'.. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS R. T. Rockafellar from the MICHIGAN MATHEMATICAL vol. 16 (1969) pp. 397-407 JOURNAL LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERATORS
More informationMath 334 A1 Homework 3 (Due Nov. 5 5pm)
Math 334 A1 Homework 3 Due Nov. 5 5pm No Advanced or Challenge problems will appear in homeworks. Basic Problems Problem 1. 4.1 11 Verify that the given functions are solutions of the differential equation,
More information14 EE 2402 Engineering Mathematics III Solutions to Tutorial 3 1. For n =0; 1; 2; 3; 4; 5 verify that P n (x) is a solution of Legendre's equation wit
EE 0 Engineering Mathematics III Solutions to Tutorial. For n =0; ; ; ; ; verify that P n (x) is a solution of Legendre's equation with ff = n. Solution: Recall the Legendre's equation from your text or
More informationWORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- lim
WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- INSTRUCTOR: CEZAR LUPU Problem Let < x < and x n+ = x n ( x n ), n =,, 3, Show that nx n = Putnam B3, 966 Question? Problem E 334 from the American
More informationPower Series Solutions of Ordinary Differential Equations
Power Series Solutions for Ordinary Differential Equations James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University December 4, 2017 Outline Power
More informationPolynomial Solutions of Nth Order Nonhomogeneous Differential Equations
Polynomial Solutions of th Order onhomogeneous Differential Equations Lawrence E. Levine Ray Maleh Department of Mathematical Sciences Stevens Institute of Technology Hoboken,J 07030 llevine@stevens-tech.edu
More informationMATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian.
MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian. Spanning set Let S be a subset of a vector space V. Definition. The span of the set S is the smallest subspace W V that contains S. If
More informationAS PURE MATHS REVISION NOTES
AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are
More informationKing Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)
Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements
More information6.1 Matrices. Definition: A Matrix A is a rectangular array of the form. A 11 A 12 A 1n A 21. A 2n. A m1 A m2 A mn A 22.
61 Matrices Definition: A Matrix A is a rectangular array of the form A 11 A 12 A 1n A 21 A 22 A 2n A m1 A m2 A mn The size of A is m n, where m is the number of rows and n is the number of columns The
More informationMath 250B Final Exam Review Session Spring 2015 SOLUTIONS
Math 5B Final Exam Review Session Spring 5 SOLUTIONS Problem Solve x x + y + 54te 3t and y x + 4y + 9e 3t λ SOLUTION: We have det(a λi) if and only if if and 4 λ only if λ 3λ This means that the eigenvalues
More informationChapter 2 Vector-matrix Differential Equation and Numerical Inversion of Laplace Transform
Chapter 2 Vector-matrix Differential Equation and Numerical Inversion of Laplace Transform 2.1 Vector-matrix Differential Equation A differential equation and a set of differential (simultaneous linear
More information(3) lk'll[-i,i] < ci«lkll[-i,i]> where c, is independent of n [5]. This, of course, yields the following inequality:
proceedings of the american mathematical society Volume 93, Number 1, lanuary 1985 MARKOV'S INEQUALITY FOR POLYNOMIALS WITH REAL ZEROS PETER BORWEIN1 Abstract. Markov's inequality asserts that \\p' \\
More informationAdditional Practice Lessons 2.02 and 2.03
Additional Practice Lessons 2.02 and 2.03 1. There are two numbers n that satisfy the following equations. Find both numbers. a. n(n 1) 306 b. n(n 1) 462 c. (n 1)(n) 182 2. The following function is defined
More informationEquations with regular-singular points (Sect. 5.5).
Equations with regular-singular points (Sect. 5.5). Equations with regular-singular points. s: Equations with regular-singular points. Method to find solutions. : Method to find solutions. Recall: The
More informationTHE MOTION OF FOUR RECTILINEAR VORTEX FILAMENTS BY S. K. LAKSHMANA RAO (Department of Power Engineering, Indian Institute of Science, Bangalore-3)
THE MOTION OF FOUR RECTILINEAR VORTEX FILAMENTS BY S. K. LAKSHMANA RAO (Department of Power Engineering, Indian Institute of Science, Bangalore-3) Received April 18, 1953 (Communicated by Prof. B. S. Madhava
More information12d. Regular Singular Points
October 22, 2012 12d-1 12d. Regular Singular Points We have studied solutions to the linear second order differential equations of the form P (x)y + Q(x)y + R(x)y = 0 (1) in the cases with P, Q, R real
More informationSection Taylor and Maclaurin Series
Section.0 Taylor and Maclaurin Series Ruipeng Shen Feb 5 Taylor and Maclaurin Series Main Goal: How to find a power series representation for a smooth function us assume that a smooth function has a power
More informationON THE GIBBS PHENOMENON FOR HARMONIC MEANS FU CHENG HSIANG. 1. Let a sequence of functions {fn(x)} converge to a function/(se)
ON THE GIBBS PHENOMENON FOR HARMONIC MEANS FU CHENG HSIANG 1. Let a sequence of functions {fn(x)} converge to a function/(se) for x0
More information