1. The problem states the Bessel function of order $-\frac{1}{2}$ is given by: $$J_{-\frac{1}{2}}(x) = \sqrt{\frac{2}{x \pi}} \cos x$$ 2. This formula expresses the Bessel function of the first kind for order $-\frac{1}{2}$ in terms of elementary functions: a square root and cosine. 3. The Bessel functions $J_\nu(x)$ are solutions to Bessel's differential equation and have many applications in physics and engineering. 4. Here, the formula shows that for half-integer negative order, the Bessel function can be written explicitly without infinite series. 5. No further simplification is needed as this is the closed form expression. Final answer: $$J_{-\frac{1}{2}}(x) = \sqrt{\frac{2}{x \pi}} \cos x$$