Foundations — Full Fourier series — coefficients derivation

4.7.5 · D1 · Maths › Partial Differential Equations › Full Fourier series — coefficients derivation

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Parent note mein bahut saara notation aise use hua hai jaise tum pehle se jaante ho: $f(x)$, $[-L,L]$, $\sin \frac{n\pi x}{L}$, ${\int }_{-L}^L$, "odd/even", "orthogonality", $a_0/2$. Neeche hum inme se har ek ko zero se earn karte hain, har ek pehle wale ke upar build hota hua. Ek samajhdar 12-saal-ka baccha line one se shuru kar sakta hai.

1. $x$ aur $f(x)$ — input aur output

Picture: ek graph par ek curve. Horizontal axis $x$ hai; har $x$ ke upar curve ki height $f(x)$ hai.

Yeh topic ko kyun chahiye: woh cheez jo hum rebuild karna chahte hain — ek sound wave, ek temperature profile, ek voltage — bilkul aisi hi ek curve $f(x)$ hai.

2. Period aur interval $[-L,L]$

Symbol $[-L,L]$ ka matlab sirf yeh hai: "$-L$ se $L$ tak ke saare $x$ values, endpoints included." Letter $L$ ek half-period hai: ek copy ki width ka aadha. Agar period $2\pi$ hai, toh $L=\pi$ (yeh parent mein Example 2 hai).

Picture: ek wallpaper pattern ki ek tile, baayein $-L$ se daayein $+L$ tak; poori wall woh tile baar baar stamp hokar banti hai.

Yeh topic ko kyun chahiye: hum hamesha sirf ek period study karte hain. Agar hum $f$ ko $[-L,L]$ par sahi kar lein, toh periodicity baaki sab kuch free mein fill kar deti hai.

3. Sine aur cosine as spinning height aur width

Jaise $\theta$ badhta hai, dono smooth up-and-down waves trace karte hain. Yeh sabse purest possible wiggles hain — ek hump upar, ek hump neeche, repeat.

4. Frequency: $\frac{n\pi x}{L}$ ka matlab

Hum sirf $\sin x$ use nahi karte; hum $\sin \frac{n\pi x}{L}$ use karte hain. Chalo us angle ko piece by piece unpack karte hain.

• $\frac{\pi x}{L}$: jaise $x$ $-L$ se $L$ tak run karta hai, yeh angle $-\pi$ se $\pi$ tak run karta hai — ek period mein bilkul ek full wave. Yeh sabse slow wave hai, $n=1$.
• Integer $n=1,2,3,\dots$ harmonic number hai. Angle ko $n$ se multiply karne par wave $n$ guna tez wiggle karti hai, isliye ek period mein $n$ complete waves fit hoti hain.

Yeh topic ko kyun chahiye: inme se har ek ka period $2L$ ko divide karta hai, isliye inka koi bhi sum har $2L$ mein repeat karta hai — $f$ se match karta hua. Alag $n$ = alag pitch; coefficients bataate hain har pitch kitni loud hai.

5. Summation sign $\sum _{n=1}^{\infty }$

Picture: waves ka ek stack ek doosre ke upar rakha hua, unki heights har $x$ par add hoke final curve banati hain.

Yeh topic ko kyun chahiye: Fourier series $f(x)=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }[a_n\cos \frac{n\pi x}{L}+b_n\sin \frac{n\pi x}{L}]$ yahi ek endless sum hai. $a_n,b_n$ loudness ke knobs hain.

6. Coefficients $a_0,a_n,b_n$

Socho ek graphic-equaliser: har $a_n$ aur $b_n$ ek slider setting hai.

7. Integral ${\int }_{-L}^{L}g(x)dx$ — "signed area"

Picture: curve ke neeche region ko shade karo. Axis ke upar = "+" shading, neeche = "−" shading. Integral hai (+ shading) minus (− shading).

8. Even aur odd functions — symmetry shortcut

Do facts heavy lifting karte hain:

• Ek odd function ka $[-L,L]$ par equal + aur − area hota hai, isliye ${\int }_{-L}^L(\text{odd})dx=0$.
• $\text{even}\times \text{even}=\text{even}$, $\text{odd}\times \text{odd}=\text{even}$, $\text{even}\times \text{odd}=\text{odd}$ (sign multiplication jaisa hi rule).

Yeh topic ko kyun chahiye: agar $f$ odd hai, toh kuch compute karne se pehle hi har $a_n$ zero ho jaata hai (parent mein Example 1 aur 2). Agar $f$ even hai, toh har $b_n$ zero ho jaata hai. Isliye parent kehta hai "pehle symmetry check karo." Dekho Even and odd functions.

9. Orthogonality — waves jo overlap nahi karti

10. Boxed formulas jinki taraf hum ja rahe hain

Prerequisite map

Related vault topics

Jab yeh foundations solid ho jayein, same machinery Half-range Fourier series (sine and cosine), Complex (exponential) Fourier series, Parseval's theorem, Convergence of Fourier series (Dirichlet conditions), aur Separation of variables for the heat equation ko bhi power karti hai.

Equipment checklist

Right-hand side cover karo aur khud ko test karo. Har ::: line ek self-check hai.