Because f(t) = ln t has an infinite discontinuity at t = 0 it might be assumed that ℒ{ln t} does not exist; however, this is incorrect. The point of this problem to guide you through the formal steps leading to the Laplace transform of f(t) = ln t, t > 0.

(a) Use integration by parts to show that

b) If ℒ{ln t} = Y(s), use Theorem 7.4.1 with n = 1 to show that part (a) becomes

Find an explicit solution Y(s) of the foregoing differential equation.

(c) Finally, the integral definition of Euler’s constant (sometimes called the Euler-Mascheroni constant) is γ = -∫^∞₀e^-t ln t dt, where γ = 0.5772156649 . . . . Use Y(1) = -γ in the solution in part (b) to show that

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L {In t} = s L {tIn t}