Fourier Series Calculator

🧮 Fourier Series Calculator – Online Fourier Expansion Tool

Welcome to the Fourier Series Calculator, your free and easy-to-use online tool for expanding periodic functions into their Fourier series representation.

Whether you’re a student, engineer, mathematician, or researcher in Ireland, this calculator helps you quickly compute Fourier coefficients and visualize how complex periodic signals can be represented using sine and cosine functions.

With our intuitive interface, you can input any mathematical function, define the period and number of terms, and instantly get both the numerical coefficients and the plotted Fourier approximation — all within seconds.

For more mathematical and scientific tools, visit our👉 Math & Science Calculators section.

🔍 What Is the Fourier Series?

The Fourier Series is a mathematical technique that breaks down any periodic function into a sum of simple sine and cosine waves.

In simpler terms, it lets you express complex signals as combinations of basic trigonometric components, making it one of the most powerful tools in signal processing, physics, electrical engineering, and applied mathematics.

Mathematically, a function $f(x)$ defined on the interval $[-L,L]$ can be written as:

f(x)=a02+∑n=1N[ancos⁡(nπxL)+bnsin⁡(nπxL)]f(x) = \frac{a_0}{2} + \sum_{n=1}^{N} [a_n \cos(\frac{n\pi x}{L}) + b_n \sin(\frac{n\pi x}{L})]f(x)=2a0​​+n=1∑N​[an​cos(Lnπx​)+bn​sin(Lnπx​)]

where

an=1L∫−LLf(x)cos⁡(nπxL)dx,bn=1L∫−LLf(x)sin⁡(nπxL)dxa_n = \frac{1}{L}\int_{-L}^{L} f(x)\cos(\frac{n\pi x}{L})dx,\quad b_n = \frac{1}{L}\int_{-L}^{L} f(x)\sin(\frac{n\pi x}{L})dxan​=L1​∫−LL​f(x)cos(Lnπx​)dx,bn​=L1​∫−LL​f(x)sin(Lnπx​)dx

This calculator computes these coefficients numerically and displays both the original function and the Fourier approximation for easy comparison.

You can also go through - Inverse Laplace Transform Calculator

⚙️ How to Use the Fourier Series Calculator

Follow these simple steps to use our online Fourier series expansion calculator:

1. Enter the function f(x):
   Input your mathematical expression using standard JavaScript Math syntax.
   Examples:

   • Math.sin(x)

   • Math.abs(x)

   • Math.exp(-x*x)

   • 1 + 0.5*Math.cos(2*x)

2. Set the period (T):
   The default value is $2\pi$, but you can enter any period depending on your function.

3. Choose the number of Fourier terms (N):
   Higher values give a more accurate approximation but take slightly longer to compute.

4. Select integration samples:
   This controls how finely the function is integrated (recommended: 1000 for smooth results).

5. Click “Compute Fourier Series”:
   The calculator instantly generates the coefficients a0,an,bna_0, a_n, b_na0​,an​,bn​ and plots both the original function and its Fourier approximation on a graph.

6. Analyze or adjust:
   Compare results, increase terms, or change functions to see how Fourier series behave for different shapes.

🔬 How the Calculator Works

This Fourier Series Calculator uses numerical integration (Simpson’s Rule) to evaluate the Fourier coefficients accurately for any user-defined function over a specified interval.

Here’s a simplified breakdown of what happens behind the scenes:

1. Your function $f(x)$ is read and converted into a JavaScript function.

2. The program integrates $f(x)\times \cos (n\pi x/L)$ and $f(x)\times \sin (n\pi x/L)$ numerically for each $n$.

3. It calculates the coefficients a0,an,bna_0, a_n, b_na0​,an​,bn​.

4. Finally, it reconstructs the Fourier approximation fN(x)f_N(x)fN​(x) using those coefficients and plots both curves for easy visualization.

This method is robust, works for any user-defined function, and avoids manual integration errors.

💡 Why Use the Fourier Series Calculator?

• ✅ Fast & Accurate: Computes Fourier coefficients in seconds using precise numerical methods.

• ✅ Visual Learning: Instantly see how sine and cosine waves reconstruct your function.

• ✅ Flexible Input: Works with any mathematical function, including discontinuous or nonlinear ones.

• ✅ Educational Tool: Ideal for students, teachers, and engineers studying signal decomposition.

• ✅ Free & Online: 100% web-based — no downloads or software required.

• ✅ Built for Ireland: Optimized for users and learners across Irish schools, universities, and engineering programs.

📘 Example Use Cases

• Physics: Representing oscillations and waveforms

• Electrical Engineering: Analyzing AC circuits and signal frequency components

• Mathematics Education: Demonstrating periodic function behavior

• Computer Science: Understanding digital signal processing (DSP) fundamentals

❓ Frequently Asked Questions (FAQ)

1. What is a Fourier Series Calculator?

A Fourier Series Calculator is an online tool that expands a given periodic function into a trigonometric series using sine and cosine functions. It computes the Fourier coefficients and shows both numerical results and a visual plot.

2. How do I enter functions in the calculator?

You can type any valid JavaScript-style expression using Math. syntax. For example:
Math.sin(x), Math.cos(2*x), Math.exp(-x*x), or combinations like Math.sin(x) + 0.3*Math.cos(3*x).

3. What is the period (T) in Fourier series?

The period (T) defines the interval over which your function repeats. For most trigonometric functions, $T=2\pi$. You can change it to match your function’s periodicity.

4. How many Fourier terms should I use?

For smooth functions, 10–20 terms give an accurate result. For discontinuous functions (like square or sawtooth waves), you may need more terms to reduce the Gibbs phenomenon.

5. Can I use this tool for non-periodic functions?

Yes, but the Fourier series assumes periodicity. Non-periodic functions will be treated as if they repeat over the given period $T$.