Question Number 221435 by wewji12 last updated on 05/Jun/25
$$\mathrm{g};\mathbb{R}\rightarrow\mathbb{R}\:,\:\mathrm{g}\in\mathcal{C}^{\omega} \:\mathrm{at}\:\mathbb{R}\:\mathrm{space} \\ $$$$\:\mathrm{evauate}\: \\ $$$$−\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \:{y}\centerdot\mathrm{g}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)\mathrm{d}{x}\mathrm{d}{y} \\ $$$$\mathrm{when}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{z}^{\mathrm{2}} \mathrm{g}\left({z}^{\mathrm{2}} \right)\mathrm{d}{z}=\frac{\pi}{\mathrm{2}} \\ $$