🎼 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