To determine The Laplace transformation of the given function. Explanation Formula used: The Laplace transformation of f : “The Laplace transform of f which is dented by L { f ( t ) } = F ( s ) = ∫ 0 ∞ e − s t f ( t ) d t ”. Calculation: The given function is f ( t ) = { 1 , 0 ≤ t < π 0 , π ≤ t < ∞ } . Compute the Laplace transformation of f ( t ) = { 1 , 0 ≤ t < π 0 , π ≤ t < ∞ } as follows. Substitute f ( t ) = 1 for 0 ≤ t < π and f ( t ) = 0 for π ≤ t < ∞ in F ( s ) = ∫ 0 ∞ e − s t f ( t ) d t and evaluate further. F ( s ) = ∫ 0 π e − s t ( 1 ) d t + ∫ π ∞ e − s t ( 0 )

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ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 6.1, Problem 21P

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The Laplace transformation of the given function.

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Chapter 6.1, Problem 20P

Chapter 6.1, Problem 22P

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Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2 multiple of n, i.e. n mod p, 2n mod p, ..., 2 p-1 -n mod p. Let T be the subset of S consisting of those residues which exceed p/2. Find the set T, and hence compute the Legendre symbol (7|23). The first 11 multiples of 7 reduced mod 23 are 7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8. 23 The set T is the subset of these residues exceeding 2° So T = {12, 14, 17, 19, 21}. By Gauss' lemma (Apostol Theorem 9.6), (7|23) = (−1)|T| = (−1)5 = −1. how come?

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Elementary Differential Equations

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