SUMMARY
The integral $$\int_{-\infty}^{\infty} \frac{e^{-iax} \coth[\sinh[bx]]}{\sinh[bx]} dx$$ does not converge, as confirmed by Mathematica, which indicates that the integral diverges over the interval from negative to positive infinity. The discussion suggests employing complex analysis techniques, particularly calculating the residue at the pole of order 2 located at z=0, to further analyze the integral's behavior. This approach is essential for understanding the convergence properties of integrals involving hyperbolic functions.
PREREQUISITES
- Complex analysis, specifically residue theorem
- Understanding of hyperbolic functions, particularly coth and sinh
- Familiarity with integral calculus over infinite intervals
- Experience with Mathematica for symbolic computation
NEXT STEPS
- Study the residue theorem in complex analysis
- Learn about the properties and applications of hyperbolic functions
- Explore techniques for evaluating improper integrals
- Practice using Mathematica for complex integrals and symbolic calculations
USEFUL FOR
Mathematicians, physicists, and students studying complex analysis or integral calculus, particularly those interested in evaluating improper integrals involving hyperbolic functions.