bessel_functions_posts.txt

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BESSEL FUNCTIONS - SOCIAL MEDIA POSTS
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1. TWITTER/X (< 280 chars)
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Ever wondered how drums vibrate? 🥁 Bessel functions Jₙ(x) solve x²y'' + xy' + (x² - n²)y = 0 and appear everywhere - from heat conduction to quantum mechanics!

Visualized in Python with scipy.special

#Python #Math #Physics #Science

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2. BLUESKY (< 300 chars)
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Bessel functions are solutions to cylindrical differential equations and fundamental to physics.

Explored Jₙ(x), Yₙ(x), and modified forms Iₙ(x), Kₙ(x) - verified recurrence relations and orthogonality numerically, then visualized vibrating membrane modes.

Python + SciPy make this accessible.

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3. THREADS (< 500 chars)
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Just dove deep into Bessel functions - one of math's most useful special functions!

These solve: x²y'' + xy' + (x² - n²)y = 0

Why care? They describe:
• Drum vibrations
• Electromagnetic waves in cables
• Heat flow in pipes
• Hydrogen atom wavefunctions

Built an interactive notebook computing Jₙ(x), Yₙ(x), Iₙ(x), Kₙ(x), verified orthogonality relations, and visualized circular membrane modes.

The math behind everyday physics is beautiful.

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4. MASTODON (< 500 chars)
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Created a computational notebook on Bessel functions.

Covered:
• Bessel's equation: x²y'' + xy' + (x² - n²)y = 0
• Series expansion: Jₙ(x) = Σₘ (-1)ᵐ/(m!Γ(m+n+1)) × (x/2)^(2m+n)
• Recurrence: Jₙ₋₁(x) + Jₙ₊₁(x) = (2n/x)Jₙ(x)
• Orthogonality verification via numerical integration
• Application: vibrating circular membrane modes

All verified with scipy.special. The zeros of Jₙ(x) determine natural frequencies in physical systems.

#Mathematics #Physics #Python #SciPy

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5. REDDIT (r/learnpython or r/math)
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**Title:** Visualizing Bessel Functions in Python - From Theory to Vibrating Drum Modes

**Body:**

If you've taken differential equations or physics, you've probably encountered Bessel functions but maybe found them abstract. I built a notebook that makes them tangible.

**What are Bessel functions?**

They solve this differential equation:

x²y'' + xy' + (x² - n²)y = 0

Think of it as the "circular coordinate" version of sines and cosines. Just like sin/cos appear when you solve wave equations in Cartesian coordinates, Bessel functions Jₙ(x) and Yₙ(x) appear for cylindrical symmetry.

**What the notebook covers:**

1. **Four types of Bessel functions** - Jₙ(x), Yₙ(x), Iₙ(x), Kₙ(x) with their properties
2. **Numerical verification** - Checked that recurrence relations like Jₙ₋₁(x) + Jₙ₊₁(x) = (2n/x)Jₙ(x) actually work
3. **Orthogonality** - Verified ∫₀¹ x·Jₙ(αₙₘx)·Jₙ(αₙₖx)dx = 0 for m ≠ k
4. **Physical application** - Visualized vibrating circular membrane modes (like a drum!)

**Key insight:** The zeros of Jₙ(x) determine natural frequencies. This is why drums have their characteristic sound - each mode vibrates at a frequency proportional to these zeros.

**Tools used:** numpy, scipy.special, matplotlib

Explore the full notebook: https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/bessel_functions.ipynb

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6. FACEBOOK (< 500 chars)
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Ever wonder why drums sound the way they do?

The answer involves Bessel functions - special mathematical solutions that describe how circular membranes vibrate.

Created a Python notebook exploring these fascinating functions. The coolest part: visualizing the actual vibration patterns of a drum head. Those nodal lines where it stays still? They're determined by zeros of Bessel functions!

Math isn't just abstract - it's the language describing the world around us.

Explore it yourself: https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/bessel_functions.ipynb

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7. LINKEDIN (< 1000 chars)
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Bessel Functions: Bridging Theory and Computation

Just completed a comprehensive computational analysis of Bessel functions - fundamental solutions that appear throughout engineering and physics.

**Technical Scope:**
• Implemented visualization of all four Bessel function types: Jₙ(x), Yₙ(x), Iₙ(x), Kₙ(x)
• Numerically verified recurrence relations and orthogonality properties
• Computed zeros of Bessel functions with scipy.special.jn_zeros()
• Applied theory to physical problem: vibrating circular membrane modes

**Engineering Applications:**
These functions are essential for:
- Electromagnetic waveguide design
- Heat transfer in cylindrical geometries
- Acoustic analysis
- Signal processing (FM synthesis)

**Skills Demonstrated:**
- Scientific Python (NumPy, SciPy, Matplotlib)
- Numerical integration and verification
- Mathematical physics modeling
- Technical visualization

The notebook verifies theoretical properties computationally, bridging mathematical rigor with practical implementation.

Full notebook: https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/bessel_functions.ipynb

#ScientificComputing #Python #Mathematics #Engineering #DataVisualization

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8. INSTAGRAM (< 500 chars)
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The mathematics of drums 🥁

These plots show Bessel functions - solutions that describe how circular membranes vibrate.

Top left: Jₙ(x) oscillates like damped waves
Top right: Yₙ(x) blows up at zero
Bottom left: Modified versions for different physics
Bottom right: Zeros that determine drum frequencies

The colorful polar plots? Actual vibration modes of a drum head. Red = moving up, blue = moving down.

Math makes music make sense.

#Mathematics #Physics #Python #DataViz #Science #Visualization #Drums #WavePhysics

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