EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES
|
|
- Άγνη Βαρουξής
- 6 χρόνια πριν
- Προβολές:
Transcript
1 Commun. Korean Math. Soc. (, No., pp. 1 pissn: / eissn: EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES Muhammad Arshad, Shahid Mubeen, Kottakkaran Sooppy Nisar, and Gauhar Rahman Abstract. In this present paper, our aim is to introduce an extended Wright-Bessel function Jα,q λ,γ,c (z which is established with the help of the extended beta function. Also, we investigate certain integral transforms and generalized integration formulas for the newly defined extended Wright-Bessel function Jα,q λ,γ,c (z and the obtained results are expressed in terms of Fox-Wright function. Some interesting special cases involving an extended Mittag-Leffler functions are deduced. 1. Introduction The well known Wright Bessel function (or Bessel-Maitland function, misnamed after the second name Maitland of E. M. Wright defined in [24] as: Jα(z λ ( z n (1.1 ; λ > 1, z C. Γ(α + λn + 1n! n Pathak [19] gave the generalization of (1.1 as: Jα,β(z λ ( 1 n ( z (1.2 2 α+2β+2n Γ(β + n + 1Γ(α + β + λn + 1, n where λ >, z C \ (,, α, β C. Singh et al. [22] defined another generalization of Wright-Bessel function in the following form: Jα,q λ,γ (γ qn ( z n (1.3 (z Γ(α + λn + 1n!, n where α, λ, γ C, R(λ >, R(α > 1, R(γ > and q (, 1 N and (γ 1 and (γ nq Γ(γ+nq Γ(γ is the generalized Pochhammer symbol (see [13]. Received February 8, 217; Accepted March 14, Mathematics Subject Classification. 33B2, 33C2, 33C45, 33C6, 33B15, 26A33. Key words and phrases. gamma function, generalized hypergeometric function, Oberhettinger formula, Wright-Bessel function, generalized Wright function, Mittag-Leffler function. 1 c Korean Mathematical Society
2 2 M. ARSHAD, S. MUBEEN, K. S. NISAR, AND G. RAHMAN Clearly, if q, then (1.3 reduces to (1.1. We recall the generalized hypergeometric function p F q (z is defined in [5] as: pf q (z p F q, (α 1, (α 2, (α p ; z (1.4 (β 1, (β 2, (β q (α 1 n (α 2 n (α p n z n (β 1 n (β 2 n (β q n n!, n where α i, β j C; i 1, 2,..., p, j 1, 2,..., q and b j, 1, 2,... and (z n is the Pochhammer symbols. The gamma function is defined as: (1.5 Γ(µ t µ 1 e t dt, µ C, (1.6 Γ(z + 1 zγ(z, z C, and beta function is defined as: (1.7 B(x, y t x 1 (1 t y 1 dt. For i 1, 2,..., p and j 1, 2,..., q, the Wright type hypergeometric function is defined (see [24]-[26] by the following series as: (α i, A i 1,p pψ q (z p Ψ q ; z (1.8 (β j, B j 1,q Γ(α 1 + A 1 n Γ(α p + A p n z n Γ(β 1 + B 1 n Γ(β q + B q n n!, n where β r and µ s are real positive numbers such that q p 1 + β s α r >. s1 r1 Equation (4.1 differs from the generalized hypergeometric function p F q (z defined (3.3 only by a constant multiplier. The generalized hypergeometric function p F q (z is a special case of p Ψ q (z for A i B j 1, where i 1, 2,..., p and j 1, 2,..., q: (α 1 1,... (α p (α (1.9 q pf q ; z 1 i, 1 1,p p pψ q ; z. Γ(β j (β 1,... (β q Γ(α i (β j, 1 1,q j1 Recently Sharma and Devi [21] defined extended Wright type function in the following series: (α i, A i 1,p, (γ, 1 p+1ψ q+1 (z, p p+1 Ψ q+1 ; (z, p (β j, B j 1,q, (c, 1 i1
3 (1.1 EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 3 Γ(α 1 + A 1 n Γ(α p + A p n B p (γ + n, c γz n, Γ(β 1 + B 1 n Γ(β q + B q n Γ(c γn! n where i 1, 2,..., p and j 1, 2,..., q, R(p >, R(c > R(γ > and p. In this paper, we introduce an extended Wright-Bessel (Bessel-Maitland function as defined in (1.3 as: Jα,q λ,γ,c B p (γ + nq, c γ (c nq ( z n (1.11 (z, p B(γ, c γ Γ(α + λn + 1n!, n where p, α, λ, γ, c C, R(p >, R(λ, R(α > 1, R(γ >, R(c > and B p (x, y is an extended beta function (see [3], [2]. Remark 1.1. i If we set p in (1.11, then extended Wright-Bessel function reduces to the (1.3. ii If we set p, q in (1.11, then extended Wright-Bessel function reduces to the (1.1. iii If α is replaced by α 1 and z by z, then (1.11 reduces to J λ,γ,c α 1,q ( z Eγ,q,c (z which is an extended Mittag-Leffler function (see [14]. iv If q 1 and α is replaced by α 1 and z by z, then (1.11 reduces to J λ,γ,c α 1,1 ( z Eγ,c (z which is an extended Mittag-Leffler function (see [18]. For this continuation of our study, we recall the following result of Oberhettinger [17] (1.12 ( z α 1 z + b + β z 2 + 2bz dz ( α b 2βb β Γ(2αΓ(β α, < R(α < R(β. 2 Γ(1 + α + β For various other investigation containing special function, the reader may refer to the recent work of researchers (see [1], [4], [6], [12], [15], [16]. In the investigation of various properties and relations of the function Jα,q λ,γ,c (z as defined in (1.11, we need the following well known facts. Sneddon [23] defined Beta Euler and Laplace transforms in following forms: (1.13 B(f(z; α, β where R(α > and R(β >. (1.14 respectively. L{f(z} z α 1 (1 z β 1 f(zdz, e sz f(zdz,
4 4 M. ARSHAD, S. MUBEEN, K. S. NISAR, AND G. RAHMAN Meijer [11] defined the following K-transform integral equation as: (1.15 R v {f(x; p} (px 1 2 Kv (pxf(xdx, where the parameter p C and K v (z represent a modified Bessel function of third kind defined in [11]. Varma transform (see [2] is defined by the following integral equation as: (1.16 V (f, k, m; s (sx m 1 2 Wk,m (sxf(xdx, where W k,m (z represent Wittaker function (see [9, p. 55]. 2. Integral transforms of an extended Bessel function In this section, we investigate some integral transforms such as Beta, Laplace, Varma and K-transforms of an extended Wright-Bessel function as defined in (1.11 and we express the obtained result in terms of an extended Wright-type hypergeometric function (1.1. Theorem 2.1. Let α, λ, γ, c, µ, δ, ν C, R(α 1, R(λ, R(γ, R(µ, R(c, R(ν, p, q (, 1 \ N, then the following integral transforms holds: (2.1 z µ 1 (1 z ν 1 Jα,q λ,γ,c (xz δ, pdz Γ(ν (c, q, (µ, δ, (γ, 1 ; Γ(γ. 3Ψ 3 (α + 1, λ, (µ + ν, δ, (c, 1 ; Proof. From (1.11 and (1.13, we obtain ( x, p z µ 1 (1 z ν 1 Jα,q λ,γ,c (xz δ, pdz ( z µ 1 (1 z ν 1 B p (γ + nq, c γ (c nq ( 1 n (xz δ n dz. B(γ, c γ Γ(α + λn + 1n! n Interchanging the order of integration and summation, we have n z µ 1 (1 z ν 1 J λ,γ,c (xz δ, pdz α,q B p (γ + nq, c γ (c nq ( 1 n (x n z µ+δn 1 (1 z ν 1 dz B(γ, c γ Γ(α + λn + 1n! n B p (γ + nq, c γ (c nq ( 1 n (x n Γ(µ + δγ(ν B(γ, c γ Γ(α + λn + 1n! Γ(µ + ν + δ.
5 EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 5 Γ(ν B p (γ + nq, c γ Γ(c + nq( 1 n (x n Γ(µ + δ Γ(γ Γ(c γ Γ(α + λn + 1n! Γ(µ + ν + δ, n which upon using (1.1, we get the required result. From above theorem, we observe the following two cases. Case (i. If λ δ and µ α + 1, then (2.1 reduces to the following result: (2.2 z α (1 z ν 1 J λ,γ,c (xz λ, pdz Γ(νJ δ,γ,c α,q α+ν,q(x and Case (ii. If λ δ and µ α + 1 and z (1 z, then (2.1 reduces to the following result: (2.3 z µ 1 (1 z α J λ,γ,c (x(1 z λ, pdz Γ(µJ δ,γ,c α,q α+µ,q(x. Theorem 2.2. Let α, λ, γ, c, µ, δ C, R(α 1, R(λ, R(γ, R(µ, R(c, R(s, p, q (, 1 \ N, then the following integral transforms holds: (2.4 z µ 1 e sz Jα,q λ,γ,c (xz δ, pdz s µ Γ(γ Proof. From (1.11 and (1.14, we have z µ 1 e sz J λ,γ,c (xz δ, pdz α,q z µ 1 e sz ( n 3Ψ 2 (c, q, (µ, δ, (γ, 1 ; ( x, p s. δ (α + 1, λ, (c, 1 ; B p(γ+nq,c γ (c nq( 1 n (xz δ n B(γ,c γ Γ(α+λn+1n! Interchanging the order of integration and summation, we have z µ 1 e sz J λ,γ,c (xz δ, pdz α,q dz. B p (γ + nq, c γ (c nq ( 1 n (x n z µ+δn 1 e sz dz B(γ, c γ Γ(α + λn + 1n! n B p (γ + nq, c γ (c nq ( 1 n (x n B(γ, c γ Γ(α + λn + 1n! s µ δn Γ(µ + δn n s µ B p (γ + nq, c γ Γ(c + nq( 1 n (x n Γ(µ + δn, Γ(γ Γ(c γ Γ(α + λn + 1n! n which by using (1.1, we get the required result. From above theorem, we observe the following two particular cases:
6 6 M. ARSHAD, S. MUBEEN, K. S. NISAR, AND G. RAHMAN (2.5 (i If p, q 1, µ α + 1, λ δ, then (2.4 reduces to by using the relation z α e sz J λ,γ α,1 (xzλ dz s α 1 (1 + xs λ γ, n (a nz n n! (1 z a (see [2], and (ii If p, γ q 1, µ α + 1, λ δ and x ±t, then (2.4 reduces to ( n z α e sz J λ,1 t α,1 ( zλ dz s α 1 (2.6 n s α 1 1 t ; s λ sλ α 1 s λ ± t. s λ t < 1 Theorem 2.3. Let α, λ, δ, ρ, γ, c C and R(α 1, R(γ, R(δ, R(ρ, R(c, p and q N, then the following result holds: (2.7 t δ 1 K µ (stjα,q λ,γ,c (ωt ρ, pdt (c, q, (δ ± µ, ρ, (γ, 1 ; s δ Γ(γ 4 Ψ 2 (α + 1, λ, (c, 1 ; 2δ 2 Proof. From (1.11 and (1.15, we have t δ 1 K µ (stjα,q λ,γ,c (ωt ρ dt ( t δ 1 K µ (st s λ ( ω( 2 s ρ, p B p(γ+nq,c γ (c nq( ωt ρ n B(γ,c γ Γ(α+λn+1n! n. dt. By changing the variable t z s and interchanging of sum and integration, we have ( ( z B p (γ + nq, c γ (c nq ( ω n ( z s δ 1 s n 1 K µ (z ρ B(γ, c γ Γ(α + λn + 1n! s dz n s δ B p (γ + nq, c γ (c nq ( ω s n ρ B(γ, c γ Γ(α + λn + 1n! n ( z s δ+ρn 1 K µ (zdz. Now, by using the following Mathai and Saxena ([1, p. 78] formula (2.8 x ρ 1 K µ (xdx 2 ρ 2 Γ( ρ ± µ, 2 in the above equation, we obtain ( ( z B p (γ + nq, c γ (c nq ( ω n ( z s δ 1 s n 1 K µ (z ρ B(γ, c γ Γ(α + λn + 1n! s dz n
7 EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 7 1 B p (γ + nq, c γ Γ(c + nq( ω Γ(γ s δ s n ρ Γ(c γ Γ(α + λn + 1n! 2δ ρn 2 Γ(δ + ρn ± µ n which by using (1.1, we get the required result. Theorem 2.4. Let α, λ, δ, ρ, γ, µ, νc C and R(α 1, R(γ, R(δ, R(ρ, R(c, p and q N, then the following result holds: (2.9 e st 2 t δ 1 W µ,ν (stjα,q λ,γ,c (ωt ρ, pdt (c, q, ( 1 1 s δ Γ(γ. 2 4Ψ 3 ± ν + δ, ρ, (γ, 1 ; ( ω s, p. ρ (α + 1, λ, (1 µ + δ, ρ, (c, 1 ; Proof. From (1.11 and (1.16, we have e st e st 2 t δ 1 W µ,ν (stjα,q λ,γ,c (ωt ρ dt ( 2 t δ 1 B p (γ + nq, c γ (c nq ( ωt ρ n W µ,ν (st dt. B(γ, c γ Γ(α + λn + 1n! n By changing the variable st z and interchanging of sum and integration in above equation yields e z z 2 ( s δ 1 W µ,ν (zjα,q λ,γ,c (t ρ 1 s dz s δ B p (γ + nq, c γ (c nq ( ω s n ( ρ e z z 2 ( B(γ, c γ Γ(α + λn + 1n! s δ+ρn 1 W µ,ν (zdz. n Now, by using the formula (2.1 e s 2 s β 1 W µ,ν (sds Γ( ν + βγ( 1 2 ν + β, Γ(1 µ + β (where R(β ± ν > 1 2 in above equation (see Mathai and Saxena [1, p. 79], we get e z z 2 ( s δ 1 W µ,ν (zjα,q λ,γ,c (t ρ 1 s dz B p (γ + nq, c γ Γ(c + nq( ω s n Γ( 1 ρ 2 ± ν + δ + ρn Γ(γ Γ(c γ Γ(α + λn + 1n! Γ(1 µ + δ + ρn s δ n which by using (1.1, we get the required result.
8 8 M. ARSHAD, S. MUBEEN, K. S. NISAR, AND G. RAHMAN 3. Unified integral formulas of an extended Bessel function In this section, we establish two generalized integral formulas containing extended Wright-Bessel function as defined in (1.11 which is represented in terms of Wright-type function defined in (1.1 by inserting with the suitable argument defined in (1.12. Theorem 3.1. Let α, µ, δ, v, λ, γ, c C with R(δ >, R(α > 1, R(λ >, R(γ >, R(c >, R(µ > R(δ >, p, q (, 1 N and z >, then the following result holds: ( z δ 1 z + b + ( µ z 2 + 2bz J λ,γ,c y α,q z + b + dz z 2 + 2bz Γ(2δbδ µ (µ + 1; 1, (µ δ, 1, (c, q, (γ, 1; (3.1 2 δ 1 Γ(γ 4 Ψ 4 ( y b, p. (α + 1, λ, (µ, 1, (δ + µ + 1, 1, (c, 1 Proof. Let L 1 be the left hand side of Theorem (3.1 and applying (1.11 to the integrand of (3.1, we have L 1 n z δ 1 ( z + b + z 2 + 2bz µ B p (γ + nq, c γ(c nq ( y B(γ, c γγ(λn + α + 1n! (z+b+ z 2 +2bz n dz. By interchanging the order of integration and summation, which is verified by the uniform convergence of the series under the given assumption of Theorem 3.1, we have (3.2 L 1 B p (γ + nq, c γ(c nq ( 1 n y n ( z δ 1 z + b + (µ+n z 2 + 2bz dz. B(γ, c γγ(λn + α + 1n! n By considering the assumption given in theorem 3.1, since R(µ+n > R(δ >, R(α > 1 and applying (1.12 to (3.2, we obtain L 1 B p (γ + nq, c γ(c nq ( 1 n y n 2(µ + nb (µ+n ( b Γ(2δΓ(µ + n α δ B(γ, c γγ(λn + α + 1n! 2 Γ(1 + µ + δ + n. n Applying (1.6, we get L 1 Γ(2δbδ µ ( 1 n y n B p (γ + nq, c γγ(c + nqγ(µ + n + 1Γ(µ + n δ 2 δ 1 Γ(γ Γ(c γn! b n Γ(α + λn + 1Γ(µ + nγ(δ + µ + n + 1 n which upon using (1.1, we get the required result. Theorem 3.2. Let α, µ, δ, v, λ, γ, c C with R(δ >, R(α > 1, R(λ >, R(γ >, R(c >, R(µ > R(δ >, p, q (, 1 N and z >. Then
9 EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 9 the following result holds: z δ 1 ( z + b + z 2 + 2bz ( µ J λ,γ,c yz α,q z+b+ z 2 +2bz dz (µ + 1; 1, (2δ, 2, (c, q, (γ, 1; Γ(µ δbδ µ (3.3 2 δ 1 4Ψ 4 ( y 2 Γ(γ, p. (µ, 1, (α + 1, λ, (µ + δ + 1, 2, (c, 1 Proof. Let L 2 be the left hand side of (3.2 and applying (1.11 to the integrand of (3.3, we have L 2 n z δ 1 ( z + b + z 2 + 2bz µ B p (γ + nq, c γ(c ( nq B(γ, c γγ(α + λn + 1n! yz z+b+ z 2 +2bz n dz. By interchanging the order of integration and summation, which is verified by the uniform convergence of the series under the given assumption of Theorem 3.2, we have B p (γ + nq, c γ(c nq ( 1 n y n L 2 B(γ, c γγ(α + λn + 1n! (3.4 n z δ+n 1 ( z + b + z 2 + 2bz (µ+n dz. By considering the assumption given in Theorem 3.2, since R(β > R(α >, R(α > 1 and applying (1.12 to (3.4, we obtain L 2 B p (γ + nq, c γ(c nq ( 1 n y n 2(µ + nb δ µ 1 + 2nΓ(µ δ ( Γ(2δ B(γ, c γγ(α + λn + 1n! 2δ+n Γ(1 + µ + δ + 2n n Γ(µ δbδ µ B p (γ + nq, c γ( 1 n y n Γ(c + nqγ(2δ + 2nΓ(µ + n δ 1 Γ(γ 2 n Γ(c γ Γ(α + λn + 1Γ(µ + nγ(1 + µ + δ + 2n n which upon using (1.1, we get the required result. 4. Special cases In this section, we present the extended form of generalized Mittag-Leffler functions which are the special cases of extended Wright-Bessel function defined in (1.11. Also, we prove four corollaries which are the special cases of obtained Theorems 3.1 and 3.2. Case 1. If α is replaced by α 1 and z by z, then (1.11 reduces to an extended Mittag-Leffler function defined as: (4.1 J λ,γ,c α 1,q ( z Eγ,q,c (z B p (γ + nq, c γ(c nq B(γ, c γγ(λn + αn!. n
10 1 M. ARSHAD, S. MUBEEN, K. S. NISAR, AND G. RAHMAN Case 2. If q 1 and α is replaced by α 1 and z by z, then (1.11 reduces to an extended Mittag-Leffler function (4.2 J λ,γ,c α 1,1 ( z Eγ,c (z B p (γ + n, c γ(c n B(γ, c γγ(λn + αn!. n Corollary 4.1. Let the conditions of Theorem 2.1, then the following integral transforms holds: (4.3 z µ 1 (1 z ν 1 E γ,q,c (xzδ, pdz Γ(ν (c, q, (µ, δ, (γ, 1 ; Γ(γ. 3Ψ 3 (α, λ, (µ + ν, δ, (c, 1 ; (x, p Corollary 4.2. Let the conditions of Theorem 2.1, then the following integral transforms holds: (4.4 z µ 1 (1 z ν 1 E γ,c (xzδ, pdz Γ(ν (c, 1, (µ, δ, (γ, 1 ; Γ(γ. 3Ψ 3 (α, λ, (µ + ν, δ, (c, 1 ; (x, p Corollary 4.3. Let the conditions of Theorem 2.2 are satisfied, then the following integral transforms holds: (4.5 z µ 1 e sz E γ,q,c (xzδ, pdz (c, q, (µ, δ, (γ, 1 ; Γ(γ. 3Ψ 2 (α, λ, (c, 1 ; s µ ( x s δ, p Corollary 4.4. Let the conditions of Theorem 2.2 are satisfied, then the following integral transforms holds: (4.6 z µ 1 e sz E γ,c (xzδ, pdz (c, 1, (µ, δ, (γ, 1 ; Γ(γ. 3Ψ 2 (α, λ, (c, 1 ; s µ ( x s δ, p Similarly, we can derives some corollaries by using the conditions of Theorems 2.3 and 2.4. Corollary 4.5. Assume that the conditions of Theorem 3.1 are satisfied. Then the following integral formula holds: ( z δ 1 z + b + ( µ z 2 + 2bz E γ,q,c y z + b + dz z 2 + 2bz....
11 (4.7 EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 11 (µ + 1; 1, (µ δ, 1, (c, q, (γ, 1; Γ(2δbδ µ 2 δ 1 Γ(γ 4 Ψ 4 (α, λ, (µ, 1, (δ + µ + 1, 1, (c, 1 ( y b, p Corollary 4.6. Assume that the conditions of Theorem 3.1 are satisfied. Then the following integral formula holds: ( z δ 1 z + b + ( µ z 2 + 2bz E γ,c y z + b + dz z 2 + 2bz (µ + 1; 1, (µ δ, 1, (c, 1, (γ, 1; (4.8 Γ(2δbδ µ 2 δ 1 Γ(γ 4 Ψ 4 ( y b, p. (α, λ, (µ, 1, (δ + µ + 1, 1, (c, 1 Corollary 4.7. Assume that the conditions of Theorem 3.2 are satisfied. Then the following integral formula holds: ( z δ 1 z + b + ( µ z 2 + 2bz E γ,q,c yz z + b + dz z 2 + 2bz Γ(µ δbδ µ 2 δ 1 4Ψ 4 (µ + 1; 1, (2δ, 2, (c, q, (γ, 1; (4.9 ( y 2 Γ(γ, p. (µ, 1, (α, λ, (µ + δ + 1, 2, (c, 1 Corollary 4.8. Assume that the conditions of Theorem 3.2 are satisfied. Then the following integral formula holds: ( z δ 1 z + b + ( µ z 2 + 2bz E γ,c yz z + b + dz z 2 + 2bz Γ(µ (µ + 1; 1, (2δ, 2, (c, 1, (γ, 1; δbδ µ (4.1 2 δ 1 4Ψ 4 ( y 2 Γ(γ, p. (µ, 1, (α, λ, (µ + δ + 1, 2, (c, 1 Concluding Remark. In this paper, we introduced certain integral transforms and representation formulas for newly defined extended Wright-Bessel function and express the obtained results in terms of an extended Wright-type hypergeometric function. If letting p, then we have the results of the generalized Wright-Bessel function and Mittag-Leffler functions proved in ([7], [22]. References [1] M. S. Abouzaid, A. H. Abusufian, and K. S. Nisar, Some unified integrals associated with generalized Bessel-Maitland function, Int. Bull. Math. Research 3 (216, [2] M. A. Chaudhry, A. Qadir, H. M. Srivastava, and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput. 159 (24, no. 2, [3] M. A. Chaudhry and S. M. Zubair, On a Class of Incomplete Gamma Functions with Applications, CRC Press, Boca Raton, 22. [4] J. Choi, P. Agarwal, S. Mathur, and S. D. Purohit, Certain new integral formulas involving the generalized Bessel functions, Bull. Korean Math. Soc. 51 (214, no. 4, [5] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, NewYork, Toronto, London,
12 12 M. ARSHAD, S. MUBEEN, K. S. NISAR, AND G. RAHMAN [6] S. Jain and P. Agarwal, A new class of integral relation involving general class of Polynomials and I-function, Walailak J. Sci. Tech. 12 (215, [7] N. U. Khan and T. Kashmin, Some integrals for the generalized Bessel-Maitland function, Electron. J. Math. Analy. Appl. 4 (216, no. 2, [8] V. S. Kiryakova, Generalized Fractional Calculus and Applications, Longman & J. Wiley, Harlow and New York, [9] A. M. Mathai and R. K. Saxena, On linear combinations of stochastic variables, Metrika 2 (1973, no. 3, [1] A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function Theory and Applications, Springer New York Dordrecht Heidelberg London, 21. [11] C. S. Meijer, Über eine Erweiterung der Laplace-Transform, Neder Wetensch Proc. 2 (194, [12] N. Menaria, S. D. Purohit, and R. K. Parmar, On a new class of integrals involving generalized Mittag-Leffler function, Sur. Math. Appl. 11 (216, 1 9. [13] G. M. Mitagg-Leffler, Sur la nouvelle function E α(x, C. R. Acad. Soc. Paris 137 (193, [14] E. Mittal, R. M. Pandey, and S. Joshi, On extension of Mittag-Leffler function, Appl. Appl. Math. 11 (216, no. 1, [15] K. S. Nisar and S. R. Mondal, Certain unified integral formulas involving the generalized modified k-bessel function of first kind, Commun. Korean Math. Soc. 32 (217, no. 1, [16] K. S. Nisar, R. K. Parmar, and A. H. Abusufian, Certain new unified integrals associated with the generalized K-Bessel function, Far East J. Math. Sci. 9 (216, [17] F. Oberhettinger, Tables of Mellin Transforms, Springer, New York, [18] M. A. Özarslan and B. Yilmaz, The extended Mittag-Leffler function and its properties, J. Inequal. Appl. 214 (214, 85, 1 pp. [19] R. S. Pathak, Certain convergence theorems and asymptotic properties of a generalization of Lommel and Maitland transformations, Proc. Nat. Acad. Sci. India Sect. A 36 (1966, [2] E. D. Rainville, Special Functions, The Macmillan Company, New York, 196. [21] S. C. Sharma and M. Devi, Certain Properties of Extended Wright Generalized Hypergeometric Function, Ann. Pure Appl. Math. 9 (215, [22] M. Singh, M. A. Khan, and A. H. Khan, On some properties of a generalization of Bessel-Maitland function, Inter. J. Math. Trend Tech. 14 (214, [23] I. N. Sneddon, The use of Integral Transform, Tata Mc Graw-Hill, New Delhi, [24] E. M. Wright, The asymptotic expansion of the generalized Bessel function, Proc. Lond. Math. Soc. 38 (1935, no. 2, [25], The asymptotic expansion of integral functions defined by Taylor Series, Philos. Trans. Roy. Soc. London Ser. A 238 (194, [26], The asymptotic expansion of the generalized hypergeometric function II, Proc. London. Math. Soc. 46 (1935, no. 2, Muhammad Arshad Department of Mathematics International Islamic University Islamabad, Pakistan address:
13 EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES 13 Shahid Mubeen Department of Mathematics University of Sargodha Sargodha, Pakistan address: Kottakkaran Sooppy Nisar Department of Mathematics College of Arts and Science-Wadi Aldawaser 11991, Prince Sattam bin Abdulaziz University Alkharj, Kingdom of Saudi Arabia address: Gauhar Rahman Department of Mathematics International Islamic University Islamabad, Pakistan address:
On the k-bessel Functions
International Mathematical Forum, Vol. 7, 01, no. 38, 1851-1857 On the k-bessel Functions Ruben Alejandro Cerutti Faculty of Exact Sciences National University of Nordeste. Avda. Libertad 5540 (3400) Corrientes,
Διαβάστε περισσότεραFURTHER EXTENSION OF THE GENERALIZED HURWITZ-LERCH ZETA FUNCTION OF TWO VARIABLES
FURTHER EXTENSION OF THE GENERALIZED HURWITZ-LERCH ZETA FUNCTION OF TWO VARIABLES KOTTAKKARAN SOOPPY NISAR* Abstract. The main aim of this paper is to give a new generaliation of Hurwit-Lerch Zeta function
Διαβάστε περισσότεραA summation formula ramified with hypergeometric function and involving recurrence relation
South Asian Journal of Mathematics 017, Vol. 7 ( 1): 1 4 www.sajm-online.com ISSN 51-151 RESEARCH ARTICLE A summation formula ramified with hypergeometric function and involving recurrence relation Salahuddin
Διαβάστε περισσότεραThe k-α-exponential Function
Int Journal of Math Analysis, Vol 7, 213, no 11, 535-542 The --Exponential Function Luciano L Luque and Rubén A Cerutti Faculty of Exact Sciences National University of Nordeste Av Libertad 554 34 Corrientes,
Διαβάστε περισσότεραGeneralized fractional calculus of the multiindex Bessel function
Available online at www.isr-publications.com/mns Math. Nat. Sci., 1 2017, 26 32 Research Article Journal Homepage:www.isr-publications.com/mns Generalized ractional calculus o the multiindex Bessel unction.
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραON NEGATIVE MOMENTS OF CERTAIN DISCRETE DISTRIBUTIONS
Pa J Statist 2009 Vol 25(2), 135-140 ON NEGTIVE MOMENTS OF CERTIN DISCRETE DISTRIBUTIONS Masood nwar 1 and Munir hmad 2 1 Department of Maematics, COMSTS Institute of Information Technology, Islamabad,
Διαβάστε περισσότεραM a t h e m a t i c a B a l k a n i c a. On Some Generalizations of Classical Integral Transforms. Nina Virchenko
M a t h e m a t i c a B a l k a n i c a New Series Vol. 26, 212, Fasc. 1-2 On Some Generalizations of Classical Integral Transforms Nina Virchenko Presented at 6 th International Conference TMSF 211 Using
Διαβάστε περισσότεραThe k-bessel Function of the First Kind
International Mathematical Forum, Vol. 7, 01, no. 38, 1859-186 The k-bessel Function of the First Kin Luis Guillermo Romero, Gustavo Abel Dorrego an Ruben Alejanro Cerutti Faculty of Exact Sciences National
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραReccurence Relation of Generalized Mittag Lefer Function
Palestine Journal of Mathematics Vol. 6(2)(217), 562 568 Palestine Polytechnic University-PPU 217 Reccurence Relation of Generalized Mittag Lefer Function Vana Agarwal Monika Malhotra Communicated by Ayman
Διαβάστε περισσότεραHomomorphism in Intuitionistic Fuzzy Automata
International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic
Διαβάστε περισσότεραA General Note on δ-quasi Monotone and Increasing Sequence
International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in
Διαβάστε περισσότεραDiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
Διαβάστε περισσότεραCoefficient Inequalities for a New Subclass of K-uniformly Convex Functions
International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραThe k-fractional Hilfer Derivative
Int. Journal of Math. Analysis, Vol. 7, 213, no. 11, 543-55 The -Fractional Hilfer Derivative Gustavo Abel Dorrego and Rubén A. Cerutti Faculty of Exact Sciences National University of Nordeste. Av. Libertad
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραBessel functions. ν + 1 ; 1 = 0 for k = 0, 1, 2,..., n 1. Γ( n + k + 1) = ( 1) n J n (z). Γ(n + k + 1) k!
Bessel functions The Bessel function J ν (z of the first kind of order ν is defined by J ν (z ( (z/ν ν Γ(ν + F ν + ; z 4 ( k k ( Γ(ν + k + k! For ν this is a solution of the Bessel differential equation
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραarxiv: v1 [math.ca] 2 Apr 2016
Unified Bessel, Modified Bessel, Spherical Bessel and Bessel-Clifford Functions Banu Yılmaz YAŞAR and Mehmet Ali ÖZARSLAN Eastern Mediterranean University, Department of Mathematics Gazimagusa, TRNC, Mersin,
Διαβάστε περισσότεραPROPERTIES OF CERTAIN INTEGRAL OPERATORS. a n z n (1.1)
GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 995, 535-545 PROPERTIES OF CERTAIN INTEGRAL OPERATORS SHIGEYOSHI OWA Abstract. Two integral operators P α and Q α for analytic functions in the open unit disk
Διαβάστε περισσότεραOn Generating Relations of Some Triple. Hypergeometric Functions
It. Joural of Math. Aalysis, Vol. 5,, o., 5 - O Geeratig Relatios of Some Triple Hypergeometric Fuctios Fadhle B. F. Mohse ad Gamal A. Qashash Departmet of Mathematics, Faculty of Educatio Zigibar Ade
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραThe k-mittag-leffler Function
Int. J. Contemp. Math. Sciences, Vol. 7, 212, no. 15, 75-716 The -Mittag-Leffler Function Gustavo Abel Dorrego and Ruben Alejandro Cerutti Faculty of Exact Sciences National University of Nordeste. Avda.
Διαβάστε περισσότεραRoman Witu la 1. Let ξ = exp(i2π/5). Then, the following formulas hold true [6]:
Novi Sad J. Math. Vol. 43 No. 1 013 9- δ-fibonacci NUMBERS PART II Roman Witu la 1 Abstract. This is a continuation of paper [6]. We study fundamental properties applications of the so called δ-fibonacci
Διαβάστε περισσότεραOn a Subclass of k-uniformly Convex Functions with Negative Coefficients
International Mathematical Forum, 1, 2006, no. 34, 1677-1689 On a Subclass of k-uniformly Convex Functions with Negative Coefficients T. N. SHANMUGAM Department of Mathematics Anna University, Chennai-600
Διαβάστε περισσότεραEvaluation of some non-elementary integrals of sine, cosine and exponential integrals type
Noname manuscript No. will be inserted by the editor Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type Victor Nijimbere Received: date / Accepted: date Abstract
Διαβάστε περισσότεραg-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King
Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i
Διαβάστε περισσότεραON INTEGRAL MEANS FOR FRACTIONAL CALCULUS OPERATORS OF MULTIVALENT FUNCTIONS. S. Sümer Eker 1, H. Özlem Güney 2, Shigeyoshi Owa 3
ON INTEGRAL MEANS FOR FRACTIONAL CALCULUS OPERATORS OF MULTIVALENT FUNCTIONS S. Sümer Eker 1, H. Özlem Güney 2, Shigeyoshi Owa 3 Dedicated to Professor Megumi Saigo, on the occasion of his 7th birthday
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραCommutative Monoids in Intuitionistic Fuzzy Sets
Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραMINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS
MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS FUMIE NAKAOKA AND NOBUYUKI ODA Received 20 December 2005; Revised 28 May 2006; Accepted 6 August 2006 Some properties of minimal closed sets and maximal closed
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραSPECIAL FUNCTIONS and POLYNOMIALS
SPECIAL FUNCTIONS and POLYNOMIALS Gerard t Hooft Stefan Nobbenhuis Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 8.195
Διαβάστε περισσότεραSolution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Διαβάστε περισσότεραJ. of Math. (PRC) Banach, , X = N(T ) R(T + ), Y = R(T ) N(T + ). Vol. 37 ( 2017 ) No. 5
Vol. 37 ( 2017 ) No. 5 J. of Math. (PRC) 1,2, 1, 1 (1., 225002) (2., 225009) :. I +AT +, T + = T + (I +AT + ) 1, T +. Banach Hilbert Moore-Penrose.. : ; ; Moore-Penrose ; ; MR(2010) : 47L05; 46A32 : O177.2
Διαβάστε περισσότεραFractional Calculus of a Class of Univalent Functions With Negative Coefficients Defined By Hadamard Product With Rafid -Operator
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 4, No. 2, 2, 62-73 ISSN 37-5543 www.ejpam.com Fractional Calculus of a Class of Univalent Functions With Negative Coefficients Defined By Hadamard
Διαβάστε περισσότεραSOME PROPERTIES OF FUZZY REAL NUMBERS
Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραThe Negative Neumann Eigenvalues of Second Order Differential Equation with Two Turning Points
Applied Mathematical Sciences, Vol. 3, 009, no., 6-66 The Negative Neumann Eigenvalues of Second Order Differential Equation with Two Turning Points A. Neamaty and E. A. Sazgar Department of Mathematics,
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραGÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)
IFSCOM016 1 Proceeding Book No. 1 pp. 155-161 (016) ISBN: 978-975-6900-54-3 SOME RESULTS ON S α,β AND T α,β INTUITIONISTIC FUZZY MODAL OPERATORS GÖKHAN ÇUVALCIOĞLU, KRASSIMIR T. ATANASSOV, AND SINEM TARSUSLU(YILMAZ)
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραforms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραA new modal operator over intuitionistic fuzzy sets
1 st Int. Workshop on IFSs, Mersin, 14 Nov. 2014 Notes on Intuitionistic Fuzzy Sets ISSN 1310 4926 Vol. 20, 2014, No. 5, 1 8 A new modal operator over intuitionistic fuzzy sets Krassimir Atanassov 1, Gökhan
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότεραNotes on the Open Economy
Notes on the Open Econom Ben J. Heijdra Universit of Groningen April 24 Introduction In this note we stud the two-countr model of Table.4 in more detail. restated here for convenience. The model is Table.4.
Διαβάστε περισσότεραGenerating Set of the Complete Semigroups of Binary Relations
Applied Mathematics 06 7 98-07 Published Online January 06 in SciRes http://wwwscirporg/journal/am http://dxdoiorg/036/am067009 Generating Set of the Complete Semigroups of Binary Relations Yasha iasamidze
Διαβάστε περισσότεραNotations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation. Mathematica StandardForm notation
KelvinKei Notations Traditional name Kelvin function of the second kind Traditional notation kei Mathematica StandardForm notation KelvinKei Primary definition 03.5.0.000.0 kei kei 0 Specific values Values
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραSubclass of Univalent Functions with Negative Coefficients and Starlike with Respect to Symmetric and Conjugate Points
Applied Mathematical Sciences, Vol. 2, 2008, no. 35, 1739-1748 Subclass of Univalent Functions with Negative Coefficients and Starlike with Respect to Symmetric and Conjugate Points S. M. Khairnar and
Διαβάστε περισσότεραA Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
Διαβάστε περισσότεραIntuitionistic Fuzzy Ideals of Near Rings
International Mathematical Forum, Vol. 7, 202, no. 6, 769-776 Intuitionistic Fuzzy Ideals of Near Rings P. K. Sharma P.G. Department of Mathematics D.A.V. College Jalandhar city, Punjab, India pksharma@davjalandhar.com
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότερα1. Introduction and Preliminaries.
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 22:1 (2008), 97 106 ON δ SETS IN γ SPACES V. Renuka Devi and D. Sivaraj Abstract We
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραAsymptotic Formulas for Generalized Elliptic-type Integrals
Asymptotic Formulas for Generalized Elliptic-type Integrals S.L. KALLA and VU KIM TUAN Department of Mathematics and Computer Science Faculty of Science, Kuwait University P.O. Box 5969, Safat 13060, Kuwait
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραn=2 In the present paper, we introduce and investigate the following two more generalized
MATEMATIQKI VESNIK 59 (007), 65 73 UDK 517.54 originalni nauqni rad research paper SOME SUBCLASSES OF CLOSE-TO-CONVEX AND QUASI-CONVEX FUNCTIONS Zhi-Gang Wang Abstract. In the present paper, the author
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραMATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραBoundedness of Some Pseudodifferential Operators on Bessel-Sobolev Space 1
M a t h e m a t i c a B a l k a n i c a New Series Vol. 2, 26, Fasc. 3-4 Boundedness of Some Pseudodifferential Operators on Bessel-Sobolev Space 1 Miloud Assal a, Douadi Drihem b, Madani Moussai b Presented
Διαβάστε περισσότεραAdditional Results for the Pareto/NBD Model
Additional Results for the Pareto/NBD Model Peter S. Fader www.petefader.com Bruce G. S. Hardie www.brucehardie.com January 24 Abstract This note derives expressions for i) the raw moments of the posterior
Διαβάστε περισσότεραSOLVING CUBICS AND QUARTICS BY RADICALS
SOLVING CUBICS AND QUARTICS BY RADICALS The purpose of this handout is to record the classical formulas expressing the roots of degree three and degree four polynomials in terms of radicals. We begin with
Διαβάστε περισσότεραDERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C
DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C By Tom Irvine Email: tomirvine@aol.com August 6, 8 Introduction The obective is to derive a Miles equation which gives the overall response
Διαβάστε περισσότεραHomomorphism of Intuitionistic Fuzzy Groups
International Mathematical Forum, Vol. 6, 20, no. 64, 369-378 Homomorphism o Intuitionistic Fuzz Groups P. K. Sharma Department o Mathematics, D..V. College Jalandhar Cit, Punjab, India pksharma@davjalandhar.com
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραOn Inclusion Relation of Absolute Summability
It. J. Cotemp. Math. Scieces, Vol. 5, 2010, o. 53, 2641-2646 O Iclusio Relatio of Absolute Summability Aradhaa Dutt Jauhari A/66 Suresh Sharma Nagar Bareilly UP) Idia-243006 aditya jauhari@rediffmail.com
Διαβάστε περισσότεραNormalization of the generalized K Mittag-Leffler function and ratio to its sequence of partial sums
Normalization of the generalized K Mittag-Leffler function ratio to its sequence of partial sums H. Rehman, M. Darus J. Salah Abstract. In this article we introduce an operator L k,α (β, δ)(f)(z) associated
Διαβάστε περισσότεραThe Spiral of Theodorus, Numerical Analysis, and Special Functions
Theo p. / The Spiral of Theodorus, Numerical Analysis, and Special Functions Walter Gautschi wxg@cs.purdue.edu Purdue University Theo p. 2/ Theodorus of ca. 46 399 B.C. Theo p. 3/ spiral of Theodorus 6
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
Διαβάστε περισσότεραLecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
Διαβάστε περισσότεραMeromorphically Starlike Functions at Infinity
Punjab University Journal of Mathematics (ISSN 1016-2526) Vol. 48(2)(2016) pp. 11-18 Meromorphically Starlike Functions at Infinity Imran Faisal Department of Mathematics, University of Education, Lahore,
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραGeodesic Equations for the Wormhole Metric
Geodesic Equations for the Wormhole Metric Dr R Herman Physics & Physical Oceanography, UNCW February 14, 2018 The Wormhole Metric Morris and Thorne wormhole metric: [M S Morris, K S Thorne, Wormholes
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραSTRONG DIFFERENTIAL SUBORDINATIONS FOR HIGHER-ORDER DERIVATIVES OF MULTIVALENT ANALYTIC FUNCTIONS DEFINED BY LINEAR OPERATOR
Khayyam J. Math. 3 217, no. 2, 16 171 DOI: 1.2234/kjm.217.5396 STRONG DIFFERENTIA SUBORDINATIONS FOR HIGHER-ORDER DERIVATIVES OF MUTIVAENT ANAYTIC FUNCTIONS DEFINED BY INEAR OPERATOR ABBAS KAREEM WANAS
Διαβάστε περισσότεραExpIntegralE. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
ExpIntegralE Notations Traditional name Exponential integral E Traditional notation E Mathematica StandardForm notation ExpIntegralE, Primary definition 06.34.0.000.0 E t t t ; Re 0 Specific values Specialied
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότερα