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Consider the meromorphic functionClose the contour with a large semicircle in the upper half-plane, indenting above the three real pole $-1,0,1$. The large-arc integral vanishes by Jordan's lemma, while the upper-half-plane indentation rule givesThe three residue areTheir sum is $\cos 1-1$, so the residue theorem yields

The real part of the integrand is an odd function, so its full-line principal value is zero. Its imaginary part,is an even function. Taking imaginary parts and halving the full-line integral therefore gives

Expand the exponent in the Inverse Gaussian distribution with unit shape parameter:Consequently its probability density function can be written aswhereIndeed, $-A(\eta )=\sqrt{-2\eta }=1/\mu$. This is the canonical form of a one-parameter exponential family, and its natural parameter is therefore

Here the natural statistic is $T(X)=X$. The exponential-family derivative identities giveandThus

For the given effective enumeration of partial computable functions, the diagonal halting set is

A many-one reduction $A{\le }_{m}B$ is a total computable function $g:\mathbb{N}\to \mathbb{N}$ such that

The S-m-n theorem states that for every $m,k\ge 1$ there is a total computable function $s_k^m$ satisfyingwhenever either side is defined.

Suppose first that $X$ is recursively enumerable. Choose a program that halts exactly on inputs in $X$, and let $e$ index a two-variable program that, on $(x,y)$, ignores $y$ and runs that program on $x$. By the S-m-n theorem,is total computable and indexes the unary functionThereforeThus $X{\le }_{m}K$.

Conversely, suppose $X{\le }_{m}K$ through a total computable $g$. On input $x$, compute $g(x)$ and simulate machine $g(x)$ on its own code. This procedure halts exactly when $g(x)\in K$, equivalently exactly when $x\in X$. Hence $X$ is recursively enumerable. We have proved the many-one completeness of the halting problem:

Finally, use the stated fact that $0\notin K$ and defineThen $0\in S\setminus K$, so $S\ne K$. The total computable map $n\mapsto 2n+1$ satisfiesand therefore $K{\le }_{m}S$. This is a distinct representative of the halting many-one degree.