New Dispersion Relations for Pion-Nucleon Scattering

New relations expressing the imaginary part of the scattering amplitude as a coupling constant term and an integral over the physical spectrum of the real part of the amplitude are developed. These relations are applied to compute the meson-nucleon coupling constant from the experimental phase shifts, yielding the value $f^2=0.084\mathrm{}$, and are used to compute the $s$-wave scattering lengths from the $p$-wave data.