- #1
outhsakotad
- 32
- 0
Homework Statement
I'm having difficulties with the integral sinh(ax)/sinh(pi*x) from -inf to inf.
A contour integral with sinh in the numerator and denominator is an integral that involves the hyperbolic sine function in both the top and bottom of the integrand. It can be represented as ∫(sinh(z)/sinh(z))dz, where z is a complex variable.
Contour integrals with sinh in the numerator and denominator have various applications in mathematics and physics. They are commonly used in the evaluation of complex integrals, solving differential equations, and calculating residues in complex analysis.
To evaluate a contour integral with sinh in the numerator and denominator, you can use techniques such as the residue theorem, Cauchy's integral formula, or the Cauchy-Riemann equations. It is important to choose an appropriate contour and use the properties of the hyperbolic sine function to simplify the integral.
Yes, there are special cases of contour integrals with sinh in the numerator and denominator, such as when the contour encloses poles or branch points of the integrand. These cases require special techniques to evaluate the integral, such as the method of steepest descent or the method of residues.
Yes, a contour integral with sinh in the numerator and denominator can be converted into a real integral by using techniques such as the substitution method or the parametrization method. However, these methods may not always yield a closed-form solution and may require numerical approximations.