Foundations — Half-range sine and cosine series

4.7.6 · D1 · Maths › Partial Differential Equations › Half-range sine and cosine series

Is page par kuch bhi assume nahi kiya gaya hai. Agar parent note mein koi symbol use hua hai, toh hum use yahan se ground up par build karte hain, ek aisi order mein jahan har idea pichle idea par tikta hai.

1. Function $f(x)$ kya hoti hai?

Picture ek curve hai grid par. Horizontal axis input $x$ hai; vertical axis output height $f(x)$ hai.

Topic ko yeh kyun chahiye. Poora subject ek function ko (jaise, ek rod mein temperature) simpler functions ke sum ke roop mein rewrite karne ke baare mein hai (waves). Tum kisi cheez ko rewrite nahi kar sakte jab tak tum exactly nahi jaante woh cheez kya hai: har $x$ par ek height.

2. Interval $[0,L]$ aur length symbol $L$

Socho ek ruler $0$ se shuru hoke rakha gaya hai. Uska right end $L$ ke mark par baitha hai. Parent note jo kuch bhi karta hai woh in do marks ke beech hota hai.

3. Sine aur Cosine curves

Isse pehle ki hum sines aur cosines se kuch build karein, humein unhe dekhna hoga.

Stretched versions $\sin \frac{n\pi x}{L}$ aur $\cos \frac{n\pi x}{L}$. Parent note plain $\sin x$ ki jagah yahi likhta hai. Bracket ke andar do cheezein ho rahi hain:

• $\frac{\pi }{L}$ factor wave ko rescale karta hai taaki woh length $\pi$ ki jagah length $L$ ke hamare interval par fit ho jaye.
• Whole number $n$ ($n=1,2,3,\dots$) count karta hai kitne humps wave $[0,L]$ ke across banati hai: $n=1$ ek gentle arch hai, $n=2$ twice as fast wiggle karta hai, aur aise hi aage.

Topic ko specifically sines aur cosines kyun chahiye. Yeh wahi shapes hain jo differentiate karne par apni form rakhti hain (sine ka derivative cosine hai, cosine ka minus sine). Kyunki heat aur wave equations mein derivatives hote hain, waves naturally woh "atoms" hain jisse solution build hota hai. Isliye Fourier ne inhe choose kiya, straight lines nahi.

4. Even aur Odd functions — mirror pictures

Yeh poore topic ka engine hai, isliye hum dheere chalenge. Dekho bhi Even and Odd Functions.

Multiplication rules (yahi half the coefficients ko khatam karta hai). Bilkul multiplication mein signs ki tarah:

5. Integral sign $\int$

$a_n$ aur $b_n$ ke formulas $\int$ se bhare hain. Yeh kya matlab hai, zero se.

Integration by parts (isliye worked examples mein yeh use hota hai). Jab tum $x\sin (nx)$ jaisi polynomial × wave integrate karte ho, koi ek single rule product ko directly handle nahi kar sakti. Integration by parts woh tool hai jo ek mushkil integral ko ek aasaan se trade karta hai polynomial ko differentiate karke ($x$ ko $1$ bana ke) jabki wave integrate karta hai. Yeh wahi elementary tool hai jo polynomial power ko shrink karta hai, jo ki yahan exactly obstacle hai.

$\int udv=uv-\int vdu.$

6. Coefficients $a_n$, $b_n$ aur sum $\sum$

Special coefficient $\frac{a_0}{2}$ $f$ ki average height hai — steady background level jo kisi bhi wave ke add hone se pehle hoti hai. Ek even function average par axis ke upar high baith sakti hai; ek odd function hamesha zero average karti hai (equal positive aur negative area), isliye sine series mein koi constant term nahi hoti.

Prerequisite map

Ise upar se neeche padho: raw ideas (function, interval, waves, symmetry, integral) do "cancellation" facts mein feed hote hain, jo coefficient formulas mein feed hote hain, jo finally parent note ki half-range series mein assemble hote hain.

Worked micro-check

Equipment checklist

Right side chhupaao aur khud test karo. Agar koi bhi line atak jaaye, parent note tackle karne se pehle upar wala matching section dobara padho.

Connections

• Parent: Half-range sine and cosine series — yahan sab kuch directly isme feed hota hai.
• Even and Odd Functions — symmetry engine poori tarah se expand kiya gaya hai.
• Fourier Series — full range — jahan coefficient formulas originally aate hain.
• Dirichlet Conditions — fine print ki series actually $f$ ke barabar kab hoti hai.