EchoKey Integral Monster Solver

🎼 EchoKey Integral Monster Solver

Using mathematical orchestration to tame the beast

∫0∞ sin(4a²/π) / [sinh(4a)sinh(2a)] da = π/16 (√2 - 1)
🎵 Step 1: Symmetry Recognition (Cyclicity)

The integrand exhibits beautiful periodic behavior in the numerator with period π in terms of 4a.

sin(4a²/π) has zeros at a = √(nπ²/4) for integer n
🎵 Step 2: Substitution (Refraction)

Let u = 2a, then du = 2da:

∫0∞ sin(2u²/π) / [sinh(2u)sinh(u)] · (1/2) du
🎵 Step 3: Partial Fractions (Synergy)

Decompose the hyperbolic functions:

1/[sinh(2u)sinh(u)] = 2/sinh(u) - 1/sinh(2u)
🎵 Step 4: Complex Contour (Outliers)

Consider f(z) = exp(2iz²/π) / sinh(z) with poles at z = nπi

Residue at πi: limz→πi (z-πi) · f(z) = -2i
Numerical Integration

Let's verify our analytical result using numerical methods:

Complex Analysis Approach

Using the residue theorem with careful contour selection:

Contour: Rectangle with vertices at ±R, ±R + πi
Poles inside: z = πi (simple pole)
                
Residue calculation:
Res(f, πi) = limz→πi (z - πi) · exp(2iz²/π) / sinh(z)
          = exp(-2πi) / cosh(πi)
          = 1 / (-1)
          = -1