Half Range Fourier Series Calculator

Calculator Inputs

This page keeps a single-column structure overall, while the input area uses 3 columns on large screens, 2 on medium screens, and 1 on mobile screens.

Half range type

Function model

Interval length L

Number of terms N

Integration steps

Evaluation point x

p1

p2

p3

p4

Decimal places

Selected model formula:

Formula Used

Half range cosine series

f(x) ≈ a0/2 + Σ a_n cos(nπx/L)

a0 = (2/L) ∫[0→L] f(x) dx

a_n = (2/L) ∫[0→L] f(x) cos(nπx/L) dx

Half range sine series

f(x) ≈ Σ b_n sin(nπx/L)

b_n = (2/L) ∫[0→L] f(x) sin(nπx/L) dx

Error metrics

Point Error = |f(x) - S_N(x)|

RMSE = √[(1/m) Σ (f(x_i) - S_N(x_i))²]

Max Error = max |f(x_i) - S_N(x_i)|

The calculator evaluates the integrals numerically with Simpson’s rule, which works well for smooth functions and controlled oscillations.

How to Use This Calculator

Choose whether you want a half range sine series or cosine series.
Select a function model that matches the target function on [0, L].
Enter the interval length L and the number of Fourier terms N.
Set the model parameters p1 to p4 according to the selected formula preview.
Choose an evaluation point inside the interval to inspect a specific approximation value.
Use higher integration steps for stronger accuracy, especially for oscillatory or rapidly growing functions.
Click the compute button to generate coefficients, comparison tables, summary metrics, and the Plotly graph.
Download CSV or PDF after calculation if you need results for assignments, reports, or revision notes.

Example Data Table

Series Type	Function Model	Example Settings	Use Case
Cosine	Linear	L = π, N = 10, p1 = 1, p2 = 0	Even extension study for `f(x)=x` on `[0, π]`.
Sine	Linear	L = 1, N = 12, p1 = 1, p2 = 0	Odd extension study with stronger endpoint behavior differences.
Cosine	Quadratic	L = 2, N = 8, p1 = 1, p2 = -2, p3 = 1	Polynomial approximation and convergence inspection across a finite interval.
Sine	Mixed	L = 3, N = 15, p1 = 0.5, p2 = 2, p3 = 3, p4 = 0	Mixed linear and oscillatory behavior for richer coefficient patterns.

Frequently Asked Questions

1. What is a half range Fourier series?

It is a Fourier expansion built from data known only on [0, L]. The calculator creates either an odd extension with sine terms or an even extension with cosine terms.

2. When should I choose the sine version?

Choose the sine series when the problem naturally matches an odd extension, such as many boundary value problems where the function value should vanish at the origin in the extended form.

3. When should I choose the cosine version?

Choose the cosine series when the problem matches an even extension or when you want a constant term and cosine harmonics to represent the function over the interval.

4. Why do more terms improve the fit?

Adding more harmonics gives the partial sum extra flexibility. That usually reduces average error, although near discontinuities or sharp changes the convergence may still be slower.

5. Why can the endpoints still look imperfect?

Endpoint behavior depends on the chosen extension and the smoothness of the original function. If the extension introduces a mismatch, oscillations or slower convergence can appear near boundaries.

6. What do the integration steps control?

They control numerical accuracy when estimating the Fourier coefficients. Smooth functions often need fewer steps, while oscillatory or steep functions benefit from larger even step counts.

7. Can I use this for coursework verification?

Yes. It is useful for checking coefficient trends, plotted behavior, and numerical approximations. For handwritten derivations, you should still present the exact analytical steps separately.

8. What does the graph show?

The graph compares the original function and the computed partial sum across the full interval. It helps you see convergence quality, oscillations, and regions with larger approximation error.