Fourier series — motivation from periodic functions: Question bank

4.7.3 · D5 · Maths › Partial Differential Equations › Fourier series — motivation from periodic functions

Jin objects par hum quiz karte hain (pehle definitions)

Neeche sab kuch ek aisi function $f$ ke baare mein hai jo repeat karti hai. Us repeat ko hum ek length se measure karte hain.

Do aur words jinpar hum rely karte hain:

Ek function ==periodic with period $T$ == hai agar $f (x + T) = f (x)$ har $x$ ke liye ho; aisa sabse chhota positive $T$ fundamental period kehlata hai.
Do waves orthogonal over one period hain agar ek period par unke product ka integral $0$ ho — ye do arrows ke right angle par milne ka function analogue hai. Neeche ki picture exactly yahi dikhati hai. Dekho Orthogonality of Functions.

Figure — Fourier series — motivation from periodic functions

Shaded area dekho: do alag waves ke liye positive humps aur negative dips cancel hokar net area zero dete hain (orthogonal); ek wave ka khud se product kabhi negative nahi hota, toh ek real positive area bachta hai — wahi bachta hua area hai jo coefficient formula read off karta hai.

True ya false — justify karo

TRUE ya FALSE: Har periodic function ka ek convergent Fourier series hota hai jo har point par uske barabar hota hai.

FALSE — tumhe Dirichlet conditions chahiye (piecewise-smooth usually sufficient case hai), aur tab bhi, jump par series midpoint

\frac{1}{2} [f (x^{+}) + f (x^{-})]

par converge karti hai, us point par

f

par nahi.

TRUE ya FALSE: Agar

f

even hai toh har

b_{n} = 0

hoga.

TRUE — ek even

f

times odd

sin (nπ x / L)

odd hota hai, aur symmetric interval

[- L, L]

par ek odd integrand integrate hokar

0

deta hai. Dekho Even and Odd Functions.

TRUE ya FALSE: Agar saare

b_{n} = 0

hain, toh

f

zaroor even hogi.

FALSE ek raw function ke liye, lekin effectively uske periodic extension ke liye TRUE — vanishing

b_{n}

kehta hai ki series (aur isliye reconstructed periodic signal) even hai. Kisi

f

ko half-interval par choose kiya ja sakta hai even (cosine series) ya odd (sine series); coefficients tumhare chosen extension ke hisaab se aate hain.

TRUE ya FALSE: Constant term

a_{0} /2

ek period par

f

ke average value ke barabar hai.

TRUE —

a_{0} = \frac{1}{L} \int_{- L}^{L} f d x

, aur us integral ko interval length

2 L

(true average) se divide karne par exactly

a_{0} /2

milta hai.

TRUE ya FALSE: Zyada Fourier terms add karne se jump ke paas maximum error hamesha chhoti hoti hai.

FALSE — jump ke paas overshoot hamesha jump height ka lagbhag

9%

rehta hai; zyada terms use sirf jump ke karib squeeze karte hain, uski height chhoti nahi karte. Ye Gibbs Phenomenon hai.

TRUE ya FALSE:

sin (2 x)

aur

sin (3 x)

[- π, π]

par orthogonal hain kyunki unki frequencies alag hain.

TRUE — base frequency ke alag integer multiples

\int_{- π}^{π} sin 2 x sin 3 x d x = 0

dete hain; product-to-sum use nonzero integer multiples ke

cos

mein badal deta hai, jo poore cycles par integrate hokar

0

dete hain.

TRUE ya FALSE:

cos (nπ x / L)

aur

cos (mπ x / L)

jab

n \neq = m

hain toh orthogonal hain, lekin do equal wale nahi hote.

TRUE —

n = m

case mein

\int cos^{2}

bachta hai, jo ek positive number (

L

) hai, yaani wave ka "length squared" nonzero hai; wahi bachta term exactly coefficient ko isolate karta hai.

TRUE ya FALSE: Ek pure cosine

cos (3 π x / L)

ka Fourier series sirf khud se bana hota hai.

TRUE — orthogonality ka matlab hai iska sirf

a_{3} = 1

coefficient nonzero hoga; recipe wahi wave return karti hai jisse tumne shuru kiya tha, poori machinery ka ek sanity check.

TRUE ya FALSE: Ek piecewise-smooth

f

jisme koi jump nahi hai, uske Fourier partial sums uniformly converge karte hain (ek saath har jagah same error bound).

TRUE — ek continuous, piecewise-smooth periodic

f

mein koi Gibbs spikes nahi hote, toh partial sums pure interval mein evenly close aate hain; precisely ek jump hi uniform convergence todta hai aur sirf pointwise convergence force karta hai.

Spot the error

Spot the error: "

f

[- L, L]

par average

\frac{1}{2 L} \int f

hai, toh

a_{0} = \frac{1}{2 L} \int_{- L}^{L} f

Average formula sahi hai, lekin woh average

a_{0}

ke barabar nahi,

a_{0} / 2

ke barabar hai. Coefficient hai

a_{0} = \frac{1}{L} \int_{- L}^{L} f

(har

a_{n}

ki tarah wahi

\frac{1}{L}

); use half karne par average milta hai.

Spot the error: "Series sines aur cosines ka ek infinite tangled sum hai, toh main ek single coefficient nahi nikal sakta."

Orthogonality use untangle karti hai: ek wave se multiply karo aur integrate karo, aur har term mar jaata hai sivaay matching wale ke. Infinite sum ek single survivor tak collapse ho jaata hai.

Spot the error: "Period

2 L

wali function ke liye, main

sin (n x)

use karunga aur coefficient

\frac{1}{π} \int

se nikaalunga."

Ye sirf tab sahi hai jab

L = π

ho. General period

2 L

ke liye wave ke andar

sin (nπ x / L)

use karna zaroor hai aur integral ke aage

\frac{1}{L}

—

L

dono jagah aana chahiye.

Spot the error: "

f

odd hai, toh

a_{n} = 0

; isliye iska Fourier series zero hai."

Odd sirf cosine coefficients

a_{n}

ko kill karta hai; sine coefficients

b_{n}

poori function carry karte hain.

a_{n}

zero karna ek shortcut hai, vanishing act nahi.

Spot the error: "Ek square wave ke jump

x = 0

par value

+ 1

aani chahiye, toh series wahan

+ 1

deti hai."

Series left aur right limits ka midpoint deti hai,

\frac{1}{2} (- 1 + 1) = 0

, regardless of how tumne

f (0)

define kiya — series jump par sirf averaged behaviour dekh sakti hai.

Spot the error: "

\int_{- L}^{L} cos (0 \cdot π x / L) cos (0 \cdot π x / L) d x = L

, baaki diagonal cases ki tarah."

n = m = 0

case special hai: integrand constant

1

hai, toh integral

2 L

hai,

L

nahi. Yahi doubled value precisely wajah hai ki constant term

a_{0} /2

kyun likha jaata hai.

Spot the error: "Kyunki sines aur cosines kabhi mix nahi hote, main

f

sin (nπ x / L)

se integrate karke

a_{n}

nikal sakta hoon."

a_{n}

isolate karne ke liye matching cosine se integrate karo; ek sine se integrate karne par

b_{n}

milega. Jo coefficient chahiye usse match karo wave ko.

Spot the error: "Complex-exponential Fourier series

f = \sum_{n = - \infty}^{\infty} c_{n} e^{inπ x / L}

sine/cosine wali se alag theory hai."

Ye same series hai repackaged karke: Euler ka

e^{i θ} = cos θ + i sin θ

har

a_{n}, b_{n}

pair ko ek single complex

c_{n}

mein fold karne deta hai (real

f

ke liye

c_{- n} = \overline{c_{n}}

ke saath). Ek theory, do notations.

Why questions

PDE solvers ko Fourier series ki zaroorat kyun padti hai?

Separation of Variables on the Heat Equation ya Wave Equation sine/cosine modes produce karta hai; ek arbitrary initial shape

f (x)

ko match karna force karta hai

f

ko exactly unhi modes ka sum likhne ke liye — wahi expansion is the Fourier series.

Constant term

a_{0} /2

kyun likha jaata hai sirf

a_{0}

ki jagah?

Taaki single formula

a_{n} = \frac{1}{L} \int f cos (nπ x / L)

n = 0

par bhi kaam kare; leftover factor of

2

(kyunki

n = 0

par

\int cos^{2}

2 L

hota hai) constant term ko

2

se divide karke absorb ho jaata hai.

Orthogonality dot product

\hat{i} \cdot \hat{j} = 0

jaisi kyun lagti hai?

Dono kehte hain "ye building blocks overlap nahi karte," toh "kitna ek hai" padhne ke liye tum uski taraf project karte ho (vectors ke liye dot, functions ke liye product integrate) aur har doosra block zero contribute karta hai.

Square wave (period

2 π

, toh

L = π

) mein sirf odd harmonics kyun aate hain?

Odd hone ki wajah se,

b_{n} = \frac{2}{π} \int_{0}^{π} sin n x d x = \frac{2}{π} \cdot \frac{1 - c o s nπ}{n} = \frac{2}{π} \cdot \frac{1 - ( - 1 ) ^{n}}{n}

; kyunki

cos nπ = (- 1)^{n}

hai, bracket even

n

ke liye

0

aur odd

n

ke liye

2

hota hai, toh saare even harmonics vanish ho jaate hain aur sirf

n = 1, 3, 5, \dots

bachte hain.

[- L, L]

par ek odd integrand integrate hokar

0

kyun deta hai?

Left half right half ka exact negative mirror hai, toh unke signed areas cancel ho jaate hain — ye har even/odd shortcut ke peeche ka engine hai.

Ek jaggy, non-smooth function perfectly smooth waves se kyun ban sakti hai?

Har finite partial sum smooth hai, lekin infinite sum ek limit hai; sharp corners sirf us limit mein emerge karte hain jahan infinitely many high-frequency waves milkar kaam karti hain — corner ke paas ye kabhi poori tarah settle nahi hote, yehi Gibbs overshoot ka source hai.

Kisi point par convergence ke liye

f

wahan continuous hone ki zaroorat kyun nahi?

Coefficients integrals hain, jo ek single point ki value se blind hain; series ek averaged picture reconstruct karti hai, toh jump par midpoint par converge karti hai bina is zaroorat ke ki

f

wahan define ho.

Fourier Transform Fourier series ke baad natural next step kyun hai?

Series ek periodic function handle karti hai (frequencies ka discrete set); jaise jaise period infinity tak badhta hai allowed frequencies ek continuum mein crowd ho jaati hain, aur sum ek integral ban jaata hai — wahi limit Fourier transform hai.

Edge cases

Edge case: Ek constant function

f (x) = c

ki Fourier series kya hai?

Sirf

a_{0} /2 = c

aur har doosra coefficient zero — ek constant apna khud ka average hai aur use kisi wave ki zaroorat nahi.

Edge case: Jump discontinuity par series exactly kya deti hai?

Left aur right limits ka midpoint

\frac{1}{2} [f (x^{+}) + f (x^{-})]

, chahe tum us point par

f

ko koi bhi value assign karo.

Edge case: Agar

f

na even hai na odd, toh kaun se coefficients bachte hain?

Generally dono

a_{n}

aur

b_{n}

— koi symmetry shortcut applicable nahi, toh tum har integral compute karte ho. Agar chaho toh

f

ko uske even part (cosines) aur odd part (sines) mein split kar sakte ho.

Edge case: Do alag piecewise functions jo sirf finitely many points par alag hain — same Fourier series?

Haan — integrals isolated points ignore karte hain, toh finitely many jagah

f

badalne se har coefficient unchanged rehta hai aur identical series produce hoti hai.

Edge case: Jaise jaise square wave ke zyada terms lete ho, corner ke paas overshoot spike ka kya hota hai?

Ye narrow hoti hai aur jump ke paas aur tightly chipakti hai lekin apni height jump ka lagbhag

9%

rakhe rehti hai — ye kabhi khatam nahi hoti, sirf discontinuity ki taraf migrate karti hai (Gibbs Phenomenon).

Edge case: Pointwise vs uniform convergence — in series ke liye distinction kya hai?

Pointwise ka matlab hai har fixed

x

par partial sums eventually target ke paas aa jaate hain, lekin kitne terms chahiye wo

x

ke saath vary kar sakta hai; uniform ka matlab hai ek term-count ek saath har jagah kaam kare. Jumps (Gibbs) sirf pointwise allow karte hain; ek continuous piecewise-smooth

f

uniform convergence deta hai.

Edge case: Real

f

ke liye, complex coefficients

c_{n}

a_{n}, b_{n}

se kya relation hai?

c_{n} = \frac{1}{2} (a_{n} - i b_{n})

jab

n \geq 1

c_{0} = a_{0} /2

, aur

c_{- n} = \overline{c_{n}}

; real coefficients ka pair aur complex conjugate pair identical information carry karte hain.

Edge case: Zero function

f (x) = 0

— iske coefficients kya hain?

Har

a_{n}

aur

b_{n}

zero hai; empty sum

0

reproduce karta hai, confirm karta hai ki recipe degenerate extreme par bhi consistent hai.

Recall Ek-line summary le jaane ke liye

Orthogonality har coefficient isolate karti hai, symmetry kaam aadha kar deti hai, aur " $=$ " sign har jagah honest hai sivaay jumps ke, jahan series politely midpoint report karti hai.