Master the essentials of Fourier Series in one clean, beginner-friendly cheat sheet. Learn the core integrals, orthogonality rules, symmetry shortcuts, and coefficient formulas that make solving Fourier problems dramatically faster. Perfect for calculus, differential equations, engineering math, and exam prep.
πΉ 1. Basic Integrals
\[ (CS.1.0) \quad \int \cos(ax)\,dx = \frac{\sin(ax)}{a} \]
\[ (CS.1.1) \quad \int \sin(ax)\,dx = -\frac{\cos(ax)}{a} \]
Assuming that for both equations \( a \neq 0 \).
πΉ 2. Orthogonality (MOST IMPORTANT)
These are the core of Fourier series. Assume that \( n,m\in \left\{ 0,1,2,\cdots \right\} \)!
Cosine Γ Cosine
\[ (CS.2.0) \quad \int_{-\pi}^{\pi} \cos(nx)\cos(mx)\,dx =
\begin{cases}
0 & n \ne m \\
\pi & n = m > 0 \\
2\pi & n = m = 0
\end{cases} \]
Sine Γ Sine
\[ (CS.2.1) \quad \int_{-\pi}^{\pi} \sin(nx)\sin(mx)\,dx =
\begin{cases}
0 & n \ne m \\
\pi & n = m > 0 \\
0 & n = m = 0
\end{cases} \]
Sine Γ Cosine
\[ (CS.2.2) \quad \int_{-\pi}^{\pi} \sin(nx)\cos(mx)\,dx = 0 \]
πΉ 3. Useful Special Cases
Constant integral
\[ (CS.3.0) \quad \int_{-\pi}^{\pi} 1\,dx = 2\pi \]
Cosine over full period
\[ (CS.3.1) \quad \int_{-\pi}^{\pi} \cos(nx)\,dx = 0 \quad (n \ne 0) \]
Sine over full period
\[ (CS.3.2) \quad \int_{-\pi}^{\pi} \sin(nx)\,dx = 0 \]
πΉ 4. Symmetry Shortcuts (Huge time saver)
Even function (f is even)
\[ (CS.4.0) \quad \int_{-\pi}^{\pi} f(x)\,dx = 2\int_{0}^{\pi} f(x)\,dx \]
Odd function (f is odd)
\[ (CS.4.1) \quad \int_{-\pi}^{\pi} f(x)\,dx = 0 \]
πΉ 5. Parity (Super Useful in Fourier Series)
- odd Γ odd = even
- even Γ even = even
- odd Γ even = odd
π Only even integrands survive over \( \left[ -\pi,\ \pi \right] \)
πΉ 6. Fourier Coefficient Formulas
Cosine coefficients
\[ (CS.6.0) \quad a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos(nx)\,dx \]
Sine coefficients
\[ (CS.6.1) \quad b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin(nx)\,dx \]
Constant coefficient
\[ (CS.6.2) \quad a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\,dx \]
The complete series
\[ (CS.6.3) \quad f(x) = \frac{a_{0}}{2}+\sum_{n=1}^{\infty}(a_{n}\cos(nx)+b_{n}\sin(nx)) \]
π§ 7. One-Page Memory Trick
- Same functions (Same nonzero frequencies)β Ο
\[ \int_{-\pi}^{\pi}\sin^2(nx)\,dx=\pi \]
\[ \int_{-\pi}^{\pi}\cos^2(nx)\,dx=\pi \]
- Different functions (frequencies differ) β 0
\[ \int_{-\pi}^{\pi}\cos(nx)\cos(mx)\,dx=0 \quad (n\neq m) \]
\[ \int_{-\pi}^{\pi}\sin(nx)\sin(mx)\,dx=0 \quad (n\neq m) \]
- Full period sine/cosine β 0
\[ \int_{-\pi}^{\pi}\sin(nx)\,dx=0 \]
\[ \int_{-\pi}^{\pi}\cos(nx)\,dx=0 \quad (n\neq0) \]
- Donβt forget symmetry β saves half the work
- If the integrand is even:
\[ \int_{-\pi}^{\pi}f(x)\,dx=2\int_0^{\pi}f(x)\,dx \] - If the integrand is odd:
\[ \int_{-\pi}^{\pi}f(x)\,dx=0 \]
- If the integrand is even:
π 8 All the formulas. All the shortcuts. One page.
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