Kinotics Physics Academy

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Karbala Colony, Udaimandir , Jodhpur

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https://t.me/kinoticsphysicsacademy

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Kinotics Physics Academy

in our page we will share entire information about csir net and gate physics . with video lectures,question series,daily practice question and notes .

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05/09/2026

📚 Modulation Property of Fourier Transform

In Fourier Transform, multiplying a signal by a complex exponential causes a shift in the frequency domain.

If

x(t) ↔ X(ω)

then,

x(t)e^(iω₀t) ↔ X(ω − ω₀)

x(t)e^(−iω₀t) ↔ X(ω + ω₀)

👉 Multiplication in the time domain = Shifting in the frequency domain.

For cosine modulation:

x(t)cos(ω₀t) ↔ ½[X(ω − ω₀) + X(ω + ω₀)]

So, cosine modulation creates two shifted copies of the original spectrum.

🎯 Important for GATE & CSIR NET Physics and Fourier Transform problems.

02/09/2026

📚 Scaling Property of Fourier Transform | GATE & CSIR NET Physics

In this video, we explain the Scaling Property of Fourier Transform in a simple and intuitive way.

If:

x(t) ↔ X(ω)

Then the scaling property is:

x(at) ↔ (1/|a|) X(ω/a)

🔹 Time compression → Frequency expansion
🔹 Time expansion → Frequency compression
🔹 Amplitude of the spectrum changes by 1/|a|
🔹 Easy tricks to remember the property
🔹 Examples: x(2t) and x(t/2)

⭐ Important for GATE Physics and CSIR NET Physics preparation.

Topics covered:
• Fourier Transform
• Scaling Property
• Time-domain scaling
• Frequency-domain scaling
• Fourier Transform properties
• GATE Physics
• CSIR NET Physics

31/08/2026

📚 Fourier Transform Properties | Complete Concept & Formula Guide | GATE & CSIR NET Physics

In this video, we explore the most important properties of the Fourier Transform that are frequently useful in solving problems for GATE Physics and CSIR NET Physics.

🔹 Linear Property
🔹 Time Shifting Property
🔹 Frequency Shifting / Modulation Property
🔹 Scaling Property
🔹 Conjugation Property
🔹 Even & Odd Function Properties
🔹 Differentiation Property
🔹 Integration Property

Each property is explained with the formula, meaning, and time-domain ↔ frequency-domain interpretation so you can understand the concept instead of just memorizing formulas.

🎯 Useful for:
GATE Physics | CSIR NET Physics | IIT-JAM | MSc Physics | Mathematical Physics

💡 Mastering Fourier Transform properties can make many numerical problems much faster and easier.

📌 Subscribe to Kinotics Physics Academy for more Physics concepts, tricks, PYQs and problem-solving videos.

29/08/2026

📚 Fourier Transform of Sine Function | Complete Explanation

In this post, we derive the Fourier Transform of the sine function step by step using Euler’s formula and the Fourier transform of exponential functions.

Let:

f(t) = sin(ω₀t)

Fourier Transform definition:

F(ω) = ∫₋∞^∞ f(t)e⁻ⁱωᵗ dt

Using Euler’s formula:

sin(ω₀t) = [eⁱω₀ᵗ − e⁻ⁱω₀ᵗ] / 2i

And:

ℱ{eⁱω₀ᵗ} = 2πδ(ω − ω₀)

ℱ{e⁻ⁱω₀ᵗ} = 2πδ(ω + ω₀)

Therefore:

F(ω) = −iπ[δ(ω − ω₀) − δ(ω + ω₀)]

💡 Key idea: A sine wave contains two frequency components, +ω₀ and −ω₀. Its Fourier Transform therefore consists of two delta-function impulses at these frequencies.

📌 Remember:

cos(ω₀t) ↔ π[δ(ω − ω₀) + δ(ω + ω₀)]

sin(ω₀t) ↔ −iπ[δ(ω − ω₀) − δ(ω + ω₀)]

Perfect for students preparing for GATE, CSIR NET, IIT JAM, JEE & Physics exams.

28/08/2026

📚 Fourier Cosine Transform — Complete Explanation

The Fourier Cosine Transform (FCT) is used to represent a function in terms of cosine waves. It is especially useful for functions defined on the positive half-axis.

🔹 Fourier Cosine Transform:

Fc(s) = √(2/π) ∫₀^∞ f(x) cos(sx) dx

🔹 Inverse Fourier Cosine Transform:

f(x) = √(2/π) ∫₀^∞ Fc(s) cos(sx) ds

💡 Why cosine only?

When a function is extended as an even function:

f(-x) = f(x)

the sine terms cancel out, leaving only cosine terms.

Therefore:

Even extension → Fourier Cosine Transform

🔹 Important standard result:

∫₀^∞ e^(-ax) cos(sx) dx = a/(a² + s²)

where a > 0.

📌 Applications:
• Heat equation
• Wave equation
• Boundary-value problems
• Signal processing
• Engineering & physics
• Diffusion and heat-conduction problems

⭐ Remember:

Fourier Cosine Transform converts a function into its cosine frequency components.

26/08/2026

Fourier Transform Explained 📊 | From Time Domain to Frequency Domain

What if a complicated signal is actually made up of many simple frequencies? 🤯

The Fourier Transform helps us find those hidden frequency components by converting a signal from the time domain into the frequency domain.

🔹 Fourier Transform:

X(f) = ∫₋∞^∞ x(t)e⁻ⁱ²πᶠᵗ dt

🔹 Inverse Fourier Transform:

x(t) = ∫₋∞^∞ X(f)eⁱ²πᶠᵗ df

For example:

x(t) = 3sin(2π100t) + 2sin(2π300t)

The Fourier Transform reveals that the signal contains 100 Hz and 300 Hz components.

📌 In simple words:
Time domain → HOW the signal changes
Frequency domain → WHAT frequencies are inside the signal

Fourier Transform is widely used in physics, engineering, audio processing, communication, optics, medical imaging, and signal processing.

25/08/2026

Parseval’s Identity in Fourier Series 📚

Parseval’s Identity connects a function with the sum of the squares of its Fourier coefficients.

Fourier Series:

f(x) = a₀/2 + Σₙ₌₁∞ [aₙ cos(nx) + bₙ sin(nx)]

Parseval’s Identity:

(1/π) ∫₋π^π |f(x)|² dx = a₀²/2 + Σₙ₌₁∞ (aₙ² + bₙ²)

For an even function:

bₙ = 0

(1/π) ∫₋π^π f²(x) dx = a₀²/2 + Σₙ₌₁∞ aₙ²

For an odd function:

a₀ = 0, aₙ = 0

(1/π) ∫₋π^π f²(x) dx = Σₙ₌₁∞ bₙ²

💡 Key Point: Parseval’s Identity is very useful for evaluating infinite series and solving problems in Fourier Series.

Useful for GATE, CSIR NET, IIT JAM & Physics/Mathematics preparation.

24/08/2026

Half Range Cosine Series | Fourier Series 📚

In this video, we explain Half Range Cosine Series step by step, with the concept of even extension, important formulas, and a solved example.

🔹 What is Half Range Cosine Series?
🔹 What is Even Extension?
🔹 Why do only cosine terms appear?
🔹 Important formulas for \(a_0\) and \(a_n\)
🔹 Step-by-step solved example
🔹 Useful tricks for GATE & CSIR NET Physics

📌 Main Formula:

f(x) = a₀/2 + Σ aₙ cos(nπx/L)

📌 Coefficient of a₀:

a₀ = (2/L) ∫₀ᴸ f(x) dx

📌 Coefficient of aₙ:

aₙ = (2/L) ∫₀ᴸ f(x) cos(nπx/L) dx

⭐ Key Concept:
Half Range Cosine Series is obtained by extending the function as an even function over the interval −L < x < L. Therefore, only cosine terms appear in the Fourier series.

🎯 Useful for GATE Physics, CSIR NET Physics, IIT JAM and Mathematical Physics preparation.

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22/08/2026

Half-Range Sine Series | Fourier Series

In this video, we learn the Half-Range Sine Series in Fourier Series from basics.

A function defined on \(0 < x < L\) is extended as an odd function over \((-L,L)\). Therefore, the Fourier series contains only sine terms.

Main Formula:

f(x) = Σ [bₙ sin(nπx/L)]

Coefficient:

bₙ = (2/L) ∫₀ᴸ f(x) sin(nπx/L) dx

Key Concept:

Half-Range Sine Series → Odd Extension → Only Sine Terms

We also solve the example:

f(x) = x, 0 < x < π

and obtain:

x = 2 Σ [(-1)ⁿ⁺¹/n] sin(nx)

This topic is important for GATE Physics, CSIR NET, IIT JAM, JEST and Mathematical Physics preparation.

21/08/2026

📚 Even & Odd Functions in Fourier Series

Understanding even and odd functions can make Fourier series problems much easier and faster. In this concept, symmetry tells us which terms will disappear automatically.

🔵 Even Function:
\(f(-x)=f(x)\)
➡️ Symmetric about the y-axis
➡️ Only cosine terms appear
➡️ \(b_n=0\)

🔴 Odd Function:
\(f(-x)=-f(x)\)
➡️ Symmetric about the origin
➡️ Only sine terms appear
➡️ \(a_0=0,\ a_n=0\)

🔥 Quick Trick:
EVEN → COSINE ONLY
ODD → SINE ONLY

This is especially useful for GATE, CSIR NET, IIT JAM and other physics/mathematics competitive exams.

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