Closed form solutions to integrals of the following type?

[eyesontheball1]

For any integral where the integrand is of the form f(θ)^z, with z a complex number, and f(θ) = sin(θ), cos(θ), tan(θ), ... etc., θ being either real or complex. Is it possible to explicitly solve for an antiderivative? I'm not aware of any such way I could use residues/series representations here, and I've tried all kinds of substitutions for integrals of this type; in particular, of [sin(θ)]^z, z being a complex constant. Thank you in advance!


David

 

[mathman]

No, even if z is real (not rational).

 

[JJacquelin]

For any integral where the integrand is of the form f(θ)^z, with z a complex number, and f(θ) = sin(θ), cos(θ), tan(θ), ... etc., θ being either real or complex. Is it possible to explicitly solve for an antiderivative? I'm not aware of any such way I could use residues/series representations here, and I've tried all kinds of substitutions for integrals of this type; in particular, of [sin(θ)]^z, z being a complex constant. Thank you in advance!


David

Hi!

most of them can be expressed in terms of hypergeometric functions.

 

[eyesontheball1]

Would you mind showing exactly how such an integral can be expressed via a hypergeometric function? Thanks in advance!

 

[JJacquelin]

For example :
http://www.wolframalpha.com/input/?i=integrate+(sin(t)^(a+i*b))*dt
http://www.wolframalpha.com/input/?i=integrate+(cos(t)^(a+i*b))*dt
http://www.wolframalpha.com/input/?i=integrate+(tan(t)^(a+i*b))*dt

 

[eyesontheball1]

Great, thank you JJacquelin!