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Closed form (or an ODE) for the integral $\int_0^\infty \frac{1+z^2}{1+z^4} \frac{z^p}{1+z^{2p}} dz$
Is there a closed form for: $$I(p)=\int_0^\infty \frac{1+z^2}{1+z^4} \frac{z^p}{1+z^{2p}} dz$$
The integral has a number of nice properties:
$$I(p)=I(-p)$$
$$I(p)=2\int_0^1 \frac{1+z^2}{1+z^4} \...
- 31.7k
3
votes
0
answers
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Is there a known transformation between $_2F_1\big(\tfrac12,\tfrac12;1;z\big)$ and $_2F_1\big(\tfrac12,\tfrac12;1;z^2\big)$?
In this post, the OP seeks a closed-form for,
$$A=\,_2F_1\big(\tfrac12,\tfrac12;1;\tfrac19\big)=1.02966\dots$$
Using the transformation,
$$\,_2F_1\big(\tfrac12,\tfrac12;1;z\big) = \tfrac2{1+\sqrt{1-z}}...
- 54.9k
3
votes
2
answers
98
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How does the recursion relation work in the solution to this differential equation (using series)?
Sorry for the vague title but it would not let me post the first step and last step of this equation (too many characters!).
How does $$\dfrac{a_0}{3n(3n-1)(3n-3)(3n-4)\cdots 9 \cdot 8 \cdot 6 \cdot ...
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