2F1(1/2,1;3/2;-z^2)=arctan(z)/z

Theorem

The following formula holds: $${}_2F_1 \left( \dfrac{1}{2}, 1; \dfrac{3}{2} ; -z^2 \right)=\dfrac{\arctan(z)}{z},$$ where ${}_2F_1$ denotes the hypergeometric 2F1 and $\arctan$ denotes the inverse tangent.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous): 15.1.5