Fourier Cosine Series Calculator

Model even extensions with accurate cosine coefficients and visual comparisons. Review errors before exporting outputs. Test custom intervals, terms, and sampling choices easily today.

Calculator Inputs

This page models a custom function on [0, L] using a finite Fourier cosine series.

Constant coefficient c0

Linear coefficient c1

Quadratic coefficient c2

Cubic coefficient c3

Quartic coefficient c4

Sine amplitude

Cosine amplitude

Angular frequency ω

Interval length L

Number of cosine terms N

Integration steps

Evaluation point x

Example Data Table

Input	Example Value	Meaning
c0	1.2	Constant part of the function
c1	-0.8	Linear x contribution
c2	0.5	Quadratic x² contribution
sin amplitude	1.1	Sine wave strength
cos amplitude	0.4	Cosine wave strength
ω	2.0	Angular frequency inside trigonometric terms
L	3.1416	Finite interval endpoint
N	8	Terms used in the partial cosine sum

Formula Used

The Fourier cosine series on 0 ≤ x ≤ L is built from cosine basis functions. This calculator evaluates coefficients numerically using Simpson integration.

f(x) ≈ S_N(x) = a₀/2 + Σ[aₙ cos(nπx/L)],  n = 1 to N

a₀ = (2/L) ∫₀ᴸ f(x) dx

aₙ = (2/L) ∫₀ᴸ f(x) cos(nπx/L) dx,  n ≥ 1

This page uses the custom model:
f(x) = c0 + c1x + c2x² + c3x³ + c4x⁴ + A sin(ωx) + B cos(ωx)

The cosine series naturally matches even extensions, so it is useful when the target function behaves well with symmetric boundary reflections.

How to Use This Calculator

Enter the polynomial coefficients for the base function.
Add optional sine and cosine amplitudes with an angular frequency.
Set the interval length L and the number of cosine terms N.
Choose Simpson steps for coefficient integration accuracy.
Enter an evaluation point x within the interval.
Press the calculate button to generate coefficients and approximation data.
Review the coefficient table, comparison table, and plotted curves.
Export the results as CSV or PDF when needed.

Frequently Asked Questions

1. What does this calculator compute?

It computes Fourier cosine coefficients, builds a partial cosine series, evaluates the approximation at a chosen point, and compares the series against the original custom function across the interval.

2. Why use a cosine series instead of a full Fourier series?

A cosine series is ideal for even extensions on finite intervals. It reduces the basis to cosine terms only and often matches boundary behavior in heat, vibration, and signal problems.

3. What function types are supported here?

This page supports a combined model with polynomial terms up to degree four plus one sine term and one cosine term. That gives flexible inputs while keeping the calculator stable and fast.

4. What does the interval length L mean?

L defines the right endpoint of the interval [0, L]. The cosine basis uses this length in every coefficient and series term, so changing L changes the entire approximation.

5. Why does the calculator use Simpson integration?

Simpson integration gives accurate numerical estimates for smooth functions with reasonable speed. It is well suited for repeated coefficient integrals and works reliably with the custom model used here.

6. What do mean error and RMS error show?

Mean absolute error shows average deviation across sampled points. RMS error emphasizes larger misses because it squares the differences before averaging and then takes the square root.

7. Does increasing the number of terms always help?

Usually it improves the approximation, especially for smooth functions. However, improvement depends on the function shape, interval, and numerical settings. Very sharp behavior may need more terms.

8. What do the CSV and PDF downloads include?

The CSV export includes coefficients and sample comparison rows. The PDF export includes summary metrics, the coefficient list, and sampled approximation data for easy reporting or sharing.

Save this file as fourier_cosine_series.php.