Show that if L{f(x)}=F(s) then L{e^ x f(x)}=F(s-k) where k is a constant. Hence find: (a) L(c^a

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Show that if L{f(x)}=F(s) then L{e^ x f(x)}=F(s-k) where k...

Show that if $L\{f(x)\}=F(s)$ then $L\left\{e^{\| x} f(x)\right\}=F(s-k)$ where $k$ is a constant.
Hence find:
(a) $L\left(c^{a x} \sin b x\right)$
(b) $L\left\{e^{\mu x} \cos b x\right)$ where $a$ and $b$ are constants in both cases.

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Key Concepts

Exponential Representation of Power Functions

Expressions like c^(ax) can be rewritten as e^(ax ln c), which allows the use of the exponential shifting theorem by identifying the exponent k as a ln c. This technique facilitates the application of the standard Laplace transform formulas to functions expressed in terms of exponential forms.

Transforms of Sine and Cosine Functions

The known Laplace transforms for sine and cosine functions, namely L{sin(bx)} = b/(s² + b²) and L{cos(bx)} = s/(s² + b²), provide crucial standard results. These are used in conjunction with the exponential shifting theorem to find the transforms of functions like c^(ax) sin(bx) and e^(?x) cos(bx).

Laplace Transform

The Laplace transform is an integral transform defined by L{f(x)} = ??? e^(–sx) f(x) dx. It converts functions of a real variable into functions of a complex variable, facilitating the analysis and solution of differential equations by transforming them into algebraic equations.

Exponential Shifting Theorem

The exponential shifting theorem states that if L{f(x)} = F(s), then L{e^(kx) f(x)} = F(s – k) for any constant k. This property is pivotal for handling functions multiplied by an exponential factor, as it shifts the s-variable in the standard Laplace transform.

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