Analyze periodic functions using numerical integration and flexible inputs. Review coefficients instantly and compare terms. Download tables for classwork, checking steps, and revision later.
| Example | Function | L | N | x | Observation |
|---|---|---|---|---|---|
| 1 | x | 3.141593 | 5 | 1 | Odd function. Most an values stay near zero. |
| 2 | abs(x) | 3.141593 | 8 | 0.5 | Even function. Most bn values stay near zero. |
| 3 | x^2 | 3.141593 | 6 | 1.2 | Smooth functions usually converge faster. |
For a function defined on the interval [-L, L], the Fourier series is written as:
f(x) ≈ a0/2 + Σ[ancos(nπx/L) + bnsin(nπx/L)]
The coefficients come from these integrals:
a0 = (1/L) ∫-LL f(x) dx
an = (1/L) ∫-LL f(x) cos(nπx/L) dx
bn = (1/L) ∫-LL f(x) sin(nπx/L) dx
This calculator evaluates those integrals numerically with Simpson’s Rule. It then builds the partial sum at your selected x value.
This calculator helps you study periodic functions with Fourier series. It finds the main integrals numerically. It also computes the Fourier coefficients. You can test a function, choose the interval, and inspect each term. The result section shows the integral, the average value, and the partial sum. This makes checking classwork easier. It also helps when algebra becomes long.
Fourier series starts with definite integrals over one full interval. Those integrals measure how much of each sine or cosine term appears in the function. The a0 value gives the average level. The an values describe cosine content. The bn values describe sine content. When a function is even, many sine terms become very small. When a function is odd, many cosine terms become very small. This pattern gives fast insight.
The page uses Simpson’s Rule for numerical integration. This method is stable for many smooth functions. You enter the number of steps, so you control the balance between speed and precision. After each coefficient is found, the calculator builds the partial sum at your chosen x value. That lets you compare the original function and the approximation. The absolute error is also shown. This is useful for revision and for practical checking.
Use this tool for homework, lecture notes, and self-study. It works well when you want a quick numerical answer. It is also useful for verifying hand calculations. The export buttons help you keep a record of coefficient tables. You can try simple inputs like x, x^2, abs(x), or sin(x). Use standard math syntax and include the multiplication sign. The calculator keeps the layout simple, clear, and easy to print.
It computes the integral over [-L, L], the Fourier coefficients a0, an, and bn, plus the partial sum at a selected x value.
You can enter standard expressions such as x, x^2, sin(x), cos(x), abs(x), sqrt(x+4), exp(x), and log(x+2). Use x as the variable.
L sets the working interval [-L, L]. The full period used by the Fourier series is 2L, so this value directly affects every coefficient.
Integration steps control how many subintervals are used by Simpson’s Rule. More steps usually improve accuracy, but they also increase calculation time.
Symmetry causes that result. Even functions often produce very small bn values. Odd functions often produce very small an values.
No. It gives numerical approximations. That is useful for checking work, exploring convergence, and studying functions that are harder to integrate by hand.
The partial sum is the truncated Fourier series using N terms. It estimates the original function at the chosen x point.
Yes. You can download a CSV file for tables and a PDF file for a clean report of the summary and coefficient values.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.