Convolution Graph Calculator

Convolution Formula:

\[ (f * g)(t) = \int f(\tau) g(t - \tau) d\tau \]

Convolution Result:

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1. What is Convolution?

Convolution is a mathematical operation on two functions that produces a third function expressing how the shape of one is modified by the other. It's widely used in signal processing, image processing, and differential equations.

2. How Does the Calculator Work?

The calculator uses the convolution integral:

\[ (f * g)(t) = \int f(\tau) g(t - \tau) d\tau \]

Where:

\( f(\tau) \) — First function
\( g(t - \tau) \) — Second function
\( \tau \) — Integration variable
\( t \) — Parameter of the resulting function

Explanation: The calculator numerically integrates the product of the two functions over the specified range to compute the convolution.

3. Applications of Convolution

Details: Convolution is fundamental in signal processing for filtering, in probability theory for sum of random variables, and in physics for describing linear systems.

4. Using the Calculator

Tips: Enter mathematical functions using standard notation (e.g., "sin(x)", "exp(-t)"). Specify appropriate integration limits for accurate results.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can I input?
A: The calculator supports basic mathematical functions including polynomials, trigonometric, exponential, and logarithmic functions.

Q2: How accurate is the numerical integration?
A: Accuracy depends on the complexity of functions and chosen integration limits. For best results, use appropriate limits that capture the significant parts of both functions.

Q3: Can I visualize the convolution result?
A: This calculator provides numerical results. For visualization, consider using graphing software with the computed convolution function.

Q4: What are common convolution pairs?
A: Common pairs include rectangular functions (resulting in triangular), Gaussian with Gaussian (resulting in Gaussian), and exponential with exponential.

Q5: Are there limitations to this calculator?
A: The calculator may struggle with highly oscillatory functions, improper integrals, or functions with discontinuities. Manual verification is recommended for critical applications.