How does cosh relate to cosine in Laplace transforms?

L{cosh(at)} = s/(s²−a²) and L{cos(at)} = s/(s²+a²) differ only by the sign in the denominator. Formally, cosh(at) = cos(jat), so substituting a → ja in the cosine transform converts +a² to −(ja)² = +a², recovering the cosh formula. Every trigonometric pair has a hyperbolic counterpart.

When do hyperbolic functions appear in system responses?

Hyperbolic functions appear in overdamped system responses (damping ratio ζ > 1) where the characteristic equation has two distinct real roots. The response e^(−αt)[A cosh(βt) + B sinh(βt)] compactly represents the sum of two decaying exponentials, common in heavily damped RLC circuits and mechanical systems.

What is the Laplace transform of sinh(at)?

L{sinh(at)} = a/(s²−a²) for Re(s) > |a|. Derived from sinh(at) = (e^(at)−e^(−at))/2 using linearity. The numerator is the constant a (like sin), while cosh has numerator s (like cos), maintaining the parallel between trigonometric and hyperbolic transform families.

Laplace of Cosh

Quick Answer

The Laplace transform of cosh(at) is L{cosh(at)} = s/(s²−a²), valid for Re(s) > |a|. Similarly, L{sinh(at)} = a/(s²−a²). These hyperbolic transform pairs mirror the trigonometric pairs but with s²−a² in the denominator instead of s²+a², reflecting real-axis poles instead of imaginary-axis poles. Compute hyperbolic transforms at www.lapcalc.com.

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Laplace of cosh(at): Formula and Derivation

The Laplace transform of cosh(at) is derived using the exponential definition cosh(at) = (e^(at) + e^(−at))/2. Applying linearity: L{cosh(at)} = (1/2)[L{e^(at)} + L{e^(−at)}] = (1/2)[1/(s−a) + 1/(s+a)] = (1/2)·2s/(s²−a²) = s/(s²−a²). The region of convergence requires Re(s) > |a|, since both exponential components must converge. The poles at s = ±a lie on the real axis, distinguishing hyperbolic transforms from trigonometric ones where poles are on the imaginary axis at s = ±ja.

Key Formulas

Laplace Transform of cosh: Comparison with Cosine

Comparing the Laplace transform of cosh(at) = s/(s²−a²) with L{cos(at)} = s/(s²+a²) reveals a structural parallel: both have numerator s, but the denominator differs by the sign of a². For cosine, s²+a² has imaginary roots ±ja (oscillation). For cosh, s²−a² has real roots ±a (exponential growth/decay). This sign difference reflects the fundamental relationship cosh(at) = cos(jat)—replacing a with ja in the cosine formula converts s²+a² to s²−(ja)² = s²+a², recovering the cosine. The formal substitution ω → ja maps every trigonometric Laplace pair to its hyperbolic counterpart.

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Damped Hyperbolic Transforms: e^(−bt)cosh(at)

Applying frequency shifting to the cosh transform gives L{e^(−bt)cosh(at)} = (s+b)/((s+b)²−a²) for Re(s) > −b+|a|. This damped hyperbolic cosine appears in overdamped system responses where the characteristic equation has two distinct real roots. For a second-order system with roots at s = −b+a and s = −b−a, the natural response involves both e^(−(b−a)t) and e^(−(b+a)t), which can be expressed as e^(−bt)[Acosh(at) + Bsinh(at)]. The hyperbolic form is particularly elegant when both roots are negative (stable overdamped), compactly representing the sum of two decaying exponentials.

Hyperbolic Functions in System Response Analysis

Hyperbolic functions arise naturally in overdamped and critically damped systems. When the discriminant of the characteristic equation is positive (ζ > 1 for second-order systems), the response involves real exponentials that can be written using cosh and sinh. For a series RLC circuit with R²/(4L²) > 1/(LC), the overdamped response is i(t) = e^(−αt)[A cosh(βt) + B sinh(βt)] where α = R/(2L) and β = √(α²−ω₀²). This representation is more compact than writing separate exponential terms and reveals the symmetry of the overdamped response around its average decay rate α.

Complete Hyperbolic Transform Table

The complete set of hyperbolic Laplace pairs includes: L{cosh(at)} = s/(s²−a²), L{sinh(at)} = a/(s²−a²), L{t·cosh(at)} = (s²+a²)/(s²−a²)², L{t·sinh(at)} = 2as/(s²−a²)², and the damped versions obtained by shifting s → s+b. The inverse transforms follow the same pattern: L⁻¹{s/(s²−a²)} = cosh(at) and L⁻¹{a/(s²−a²)} = sinh(at). These pairs, combined with partial fraction decomposition, handle any rational F(s) with real distinct poles by grouping symmetric pole pairs into hyperbolic terms. Verify all hyperbolic transforms at www.lapcalc.com.

Frequently Asked Questions

L{cosh(at)} = s/(s²−a²) for Re(s) > |a|. It is derived by writing cosh(at) = (e^(at)+e^(−at))/2 and applying linearity with the exponential transform. The poles at s = ±a are on the real axis, unlike the cosine transform which has imaginary poles.

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