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(11) Consider the Gelfand-Shilov function given by: if x > 0, Φλ (2) 10° if x ≤0. with Re(A) > 0. Show that a(z) * 03(x) = 0α+3(x). (12) Show that given the functions f(t) = t−1 and g(t) = t2-1, it holds that f(t)g(t) = +4-1B(p,q). (13) Using the Mittag-Leffler function, show that: a) E1,2(2²) = -1; b) E2.1(-z²) = cos(2); c) E2.2 (-2²) sin() = (14) Show that for the two-parameter Mittag-Leffler function, the following holds: Ea.(2) + Ea.(-2) = 2Еas (2²). (15) Compute the Laplace transform of the following Mittag-Leffler expressions: a) L{E(-λt)}; b) L{tª¹E₁(−λtº)}.

(11) Consider the Gelfand-Shilov function given by: if x > 0, Φλ (2) 10° if x ≤0. with Re(A) > 0. Show that a(z) * 03(x) = 0α+3(x). (12) Show that given the functions f(t) = t−1 and g(t) = t2-1, it holds that f(t)g(t) = +4-1B(p,q). (13) Using the Mittag-Leffler function, show that: a) E1,2(2²) = -1; b) E2.1(-z²) = cos(2); c) E2.2 (-2²) sin() = (14) Show that for the two-parameter Mittag-Leffler function, the following holds: Ea.(2) + Ea.(-2) = 2Еas (2²). (15) Compute the Laplace transform of the following Mittag-Leffler expressions: a) L{E(-λt)}; b) L{tª¹E₁(−λtº)}.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.5: Solution Of Cubic And Quartic Equations By Formulas (optional)
Problem 29E
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(11) Consider the Gelfand-Shilov function given by: if x > 0, Φλ (2) 10° if x ≤0. with Re(A) > 0. Show that a(z) * 03(x) = 0α+3(x). (12) Show that given the functions f(t) = t−1 and g(t) = t2-1, it holds that f(t)g(t) = +4-1B(p,q). (13) Using the Mittag-Leffler function, show that: a) E1,2(2²) = -1; b) E2.1(-z²) = cos(2); c) E2.2 (-2²) sin() = (14) Show that for the two-parameter Mittag-Leffler function, the following holds: Ea.(2) + Ea.(-2) = 2Еas (2²). (15) Compute the Laplace transform of the following Mittag-Leffler expressions: a) L{E(-λt)}; b) L{tª¹E₁(−λtº)}.
Transcribed Image Text:(11) Consider the Gelfand-Shilov function given by: if x > 0, Φλ (2) 10° if x ≤0. with Re(A) > 0. Show that a(z) * 03(x) = 0α+3(x). (12) Show that given the functions f(t) = t−1 and g(t) = t2-1, it holds that f(t)g(t) = +4-1B(p,q). (13) Using the Mittag-Leffler function, show that: a) E1,2(2²) = -1; b) E2.1(-z²) = cos(2); c) E2.2 (-2²) sin() = (14) Show that for the two-parameter Mittag-Leffler function, the following holds: Ea.(2) + Ea.(-2) = 2Еas (2²). (15) Compute the Laplace transform of the following Mittag-Leffler expressions: a) L{E(-λt)}; b) L{tª¹E₁(−λtº)}.
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