A new method for computing toroidal harmonics

A method for computing Legendre functions of integer order and half-odd degree is presented. The method is based on the theory of quadratic transformation of the argument, and is a generalization of Gauss’ or Landen’s transformation for computing elliptic integrals.</p

Further extensions of a Legendre function integral

The integral $$∫ z 1 ( 1 − t 2 ) β − 1 ( 1 − t 1 + t ) μ / 2 ln ⁡ ( 1 − t 2 ) P ν − 1 μ ( t ) d t \int _z^1 {{{\left ( {\frac {{1 - t}}{2}} \right )}^{\beta - 1}}{{\left ( {\frac {{1 - t}}{{1 + t}}} \right )}^{\mu /2}}\ln \left ( {\frac {{1 - t}}{2}} \right )P_{\nu - 1}^\mu (t)\;dt}$$ is evaluated as a hypergeometric function for arbitrary values of ν \nu , μ \mu , − 1 ⩽ z ⩽ 1 - 1 \leqslant z \leqslant 1 , and Re ⁡ ( β ) &gt; 0 \operatorname {Re} (\beta ) &gt; 0 .</p

More trigonometric integrals

Integrals of the form $$∫ 0 π / 2 e i p θ cos q θ d θ , ∫ 0 π / 2 e i p θ sin q θ d θ \int _0^{\pi /2} {{e^{ip\theta }}{{\cos }^q}\theta \,d\theta ,\quad \int _0^{\pi /2} {{e^{ip\theta }}{{\sin }^q}\theta \,d\theta } }$$ (p real, Re ⁡ ( q ) &gt; − 1 \operatorname {Re} (q) &gt; - 1 ) are expressed in terms of Gamma and hypergeometric functions for integer and noninteger values of q and p. The results include those of [2] as special cases.</p

On some trigonometric integrals

Expressions are obtained for the integrals $$I λ ( p ) = ∫ 0 π / 2 ( sin ⁡ λ θ sin ⁡ θ ) p d θ , J λ ( p ) = ∫ 0 π / 2 ( 1 − cos ⁡ λ θ sin ⁡ θ ) p d θ I_\lambda ^{(p)} = \int _0^{\pi /2}{\left ( {\frac {{\sin \lambda \theta }}{{\sin \theta }}} \right )^p}d\theta ,\quad J_\lambda ^{(p)} = \int _0^{\pi /2}{\left ( {\frac {{1 - \cos \lambda \theta }}{{\sin \theta }}} \right )^p}d\theta$$ for arbitrary real values of " λ \lambda ", and p = 1 , 2 p = 1,2 .</p