Exponential weighting property of Laplace transform: Multiplication by e^{-a t} in time corresponds to which operation in the s-domain?

Introduction / Context:
The frequency-shift (exponential weighting) property of the Laplace transform is a key tool for handling signals multiplied by exponentials, as in modulation, windowing, and system responses with damping or growth.

Given Data / Assumptions:

• Time-domain multiplication: x(t) e^{-a t}.
• Baseline transform: X(s) = L{x(t)}.
• We assume regions of convergence are adjusted accordingly.

Concept / Approach:

The property states L{x(t) e^{-a t}} = X(s + a). Thus multiplying by a decaying exponential shifts the transform to the right by a in the s-plane (i.e., replaces s with s + a).

Step-by-Step Solution:

Verification / Alternative check:

Test with x(t) = 1 (unit step): L{e^{-a t}} = 1/(s + a), which equals X(s + a) for X(s) = 1/s, confirming the property.

Why Other Options Are Wrong:

• Translation by −a corresponds to multiplying by e^{+a t}, not e^{−a t}.
• Multiplication by e^{-a s} in the s-domain is not a standard Laplace property for this operation.
• No change or conjugation are not applicable here.

Common Pitfalls:

• Sign confusion: e^{-a t} → X(s + a); e^{+a t} → X(s − a).
• Ignoring changes in the region of convergence after the shift.

Final Answer:

Translation by +a in s-domain: F(s + a)