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Substitution ofand integration by parts giveThus takeand require the endpoint term to vanish. Infinite contour ends must lie in sectors where $\mathrm{Re}(t^4)>0$, namely sectors centred on the positive and negative real and imaginary axes, each of angular width $\pi /4$.

Let the common final segment be the positive real ray from $0$ to $+\infty$, and choose the initial rays from $-\infty$, $+i\infty$, and $-i\infty$ to $0$. Call the resulting contours ${\gamma }_1,{\gamma }_2,{\gamma }_3$. PutSince $w_i^{(r)}(0)={\int }_{{\gamma }_i}{t}^r{e}^{-t^4/4}dt$, for $r=0,1,2$ the three rows areConsequentlyUsing the Gamma function reflection identity,The initial-data vectors are therefore linearly independent, so the three contour integrals are linearly independent solutions.

Chemical equilibrium for $p+e\leftrightarrow H$ givesand charge neutrality gives $n_p=n_e$. Insert the nonrelativistic Maxwell-Boltzmann number densities, use $m_H{c}^2=m_p{c}^2+m_e{c}^2-E_{\mathrm{bind}}$, and neglect the proton correction in the reduced mass. The spin degeneracies cancel in the ground-state approximation. One obtains Saha equationEquivalently, its reciprocal is the ionization ratio. The exponential favors neutral hydrogen as the temperature falls, while the translational phase-space factor favors ionization at high temperature.

Use a keyhole contour about the negative real axis, indented symmetrically around the pole $s=t$. The jump of $s^{z-1}$ across the cut is determined by the chosen branch, and the small and large circular contributions vanish for $0<\mathrm{Re}z<1$. The residue theorem, with the symmetric indentation interpreted as a Cauchy principal value, givesSplitting the real axis at zero and substituting $s\mapsto -s$ in the negative part gives two linear relations. Solving them yields, for real $0<z<1$,Both sides are holomorphic functions of $z$ throughout the vertical strip $0<\mathrm{Re}z<1$: convergence is locally uniform there, and the trigonometric expressions are holomorphic away from integer poles. The identity theorem therefore extends the identities from real $z$ to the whole strip.