To compute the convolution of the functions $f(t) = e^{-t}$ and $g(t) = t$, we use the formula for the convolution of two functions $f(t)$ and $g(t)$ defined as:

$(f * g)(t) = \int_0^t f(\tau) g(t - \tau) \, d\tau$

Step-by-Step Computation

We have:

• $f(t) = e^{-t}$
• $g(t) = t$

The convolution integral becomes:

$(f * g)(t) = \int_0^t e^{-\tau} (t - \tau) \, d\tau$

Now, expand $(t - \tau)$ inside the integral:

$(f * g)(t) = \int_0^t e^{-\tau} t \, d\tau - \int_0^t e^{-\tau} \tau \, d\tau$

We will now compute each term individually.

1. First Integral: $\int_0^t e^{-\tau} t \, d\tau$

Since $t$ is a constant with respect to $\tau$, we can factor it out:

$t \int_0^t e^{-\tau} \, d\tau$

The integral of $e^{-\tau}$ is:

$\int e^{-\tau} \, d\tau = -e^{-\tau}$

Evaluating from 0 to $t$:

$t \left[-e^{-\tau}\right]_0^t = t \left( -e^{-t} + e^{0} \right) = t \left( 1 - e^{-t} \right)$

2. Second Integral: $\int_0^t e^{-\tau} \tau \, d\tau$

This is a standard integral, and we can solve it using integration by parts.

Let:

• $u = \tau$ and $dv = e^{-\tau} d\tau$
• $du = d\tau$ and $v = -e^{-\tau}$

Using the integration by parts formula $\int u dv = uv - \int v du$:

$\int_0^t e^{-\tau} \tau \, d\tau = \left[ -\tau e^{-\tau} \right]_0^t + \int_0^t e^{-\tau} \, d\tau$

First, evaluate $-\tau e^{-\tau}$ from 0 to $t$:

$-\tau e^{-\tau} \Big|_0^t = -t e^{-t} + 0 = -t e^{-t}$

Now, compute the remaining integral:

$\int_0^t e^{-\tau} \, d\tau = \left[-e^{-\tau} \right]_0^t = 1 - e^{-t}$

So, the second integral is:

$-t e^{-t} + (1 - e^{-t})$

Combine the results

Now, combining the results of both integrals:

$(f * g)(t) = t(1 - e^{-t}) - \left( -t e^{-t} + 1 - e^{-t} \right)$

Simplifying:

$(f * g)(t) = t(1 - e^{-t}) + t e^{-t} - 1 + e^{-t}$

$(f * g)(t) = t - te^{-t} + t e^{-t} - 1 + e^{-t}$

The terms $- te^{-t}$ and $t e^{-t}$ cancel out, leaving:

$(f * g)(t) = t - 1 + e^{-t}$

Final Result

$(f * g)(t) = t - 1 + e^{-t}$

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Would you like more details on any part of the process? Or do you have any questions?

Here are some related questions:

1. How would you compute the convolution for different types of functions, like polynomials or sinusoidal functions?
2. What is the role of the convolution integral in signal processing?
3. How do you compute the inverse Laplace transform of a convolution?
4. Can we compute the convolution of two functions without using integration?
5. What is the relationship between convolution and the Fourier transform?

Tip: When performing convolution, always check if you can simplify the integral by factoring constants out of the integral before evaluating it.