Special Functions

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

Read the typeset paper

Read this chapter with formatted equations

1 \hypertarget{orbit-spectral-zeta-continuation-trace-zeros-and-regularized-product}{%
2 \chapter[Spectral continuation and regularization]{Orbit spectral zeta: continuation, trace zeros, and regularized
3 product}\label{app:spectral}\label{orbit-spectral-zeta-continuation-trace-zeros-and-regularized-product}}
4 
5 This chapter extends the previously defined orbit family with a self-contained continuation proof.
6 The Mittag--Leffler seed and Mellin continuation mechanism are classical.
7 
8 \hypertarget{the-completed-result}{%
9 \section{The continuation theorem}\label{the-completed-result}}
10 
11 Let \(N\ge2\) be an integer, and put \(\alpha=1/N\). Define
12 
13 \[
14 d_N(w)=\sum_{n=0}^{\infty}\frac{(-1)^n w^n}{(Nn)!}
15       =E_{N,1}(-w).
16 \]
17 
18 List its nonzero zeros \(w_j\) with multiplicity, and use the principal
19 logarithm, with \(-\pi<\operatorname{Arg}w_j<\pi\). No zero lies on the
20 omitted negative ray. The orbit zeta function
21 
22 \[
23 \mathcal Z_N(s)=\sum_j e^{-s\operatorname{Log}w_j}
24 \]
25 
26 converges absolutely exactly when \(\Re s>1/N\). It has a meromorphic
27 continuation to the whole plane, with exactly one pole: a simple pole at
28 \(s=1/N\), of residue
29 
30 \[
31 \boxed{\operatorname{Res}_{s=1/N}\mathcal Z_N(s)
32       =\frac{\sin(\pi/N)}{N\pi}.}
33 \]
34 
35 At every nonpositive integer,
36 
37 \[
38 \boxed{\mathcal Z_N(-m)=0\quad(m=0,1,2,\ldots).}
39 \]
40 
41 At the origin,
42 
43 \[
44 \boxed{\mathcal Z_N'(0)=-\log N,\qquad
45        \exp[-\mathcal Z_N'(0)]=N.}
46 \]
47 
48 The last equality is the zeta regularized product of the orbit zeros
49 under this branch convention. It is a regularized product, not
50 convergence of the ordinary product of the zeros. Positive integer values retain the exact rational coefficient law established in the preceding construction.
51 
52 \hypertarget{the-exponential-packet-and-its-exact-asymptotic}{%
53 \section{The exponential packet and its exact
54 asymptotic}\label{the-exponential-packet-and-its-exact-asymptotic}}
55 
56 Let \(\omega=e^{2\pi i/N}\). The finite root-of-unity filter gives, for
57 every \(x\in\mathbb C\),
58 
59 \[
60 F_N(x):=d_N(-x^N)
61 =\frac1N\sum_{j=0}^{N-1}e^{\omega^j x}
62 =\sum_{n=0}^{\infty}\frac{x^{Nn}}{(Nn)!}.
63 \tag{S1}
64 \]
65 
66 This equality follows by expanding the exponentials and using
67 \(N^{-1}\sum_j\omega^{jm}=1\) when \(N\mid m\), and zero otherwise. For
68 real \(x\ge0\), the power series makes \(F_N(x)>0\). Put
69 
70 \[
71 h_N(x)=\sum_{j=1}^{N-1}e^{(\omega^j-1)x},
72 \qquad c_N=1-\cos(2\pi/N)>0.
73 \]
74 
75 Then
76 
77 \[
78 F_N(x)=\frac{e^x}{N}(1+h_N(x)),\qquad
79 |h_N(x)|\le(N-1)e^{-c_Nx},\qquad
80 |h_N'(x)|\le2(N-1)e^{-c_Nx}.
81 \tag{S2}
82 \]
83 
84 Since \(1+h_N(x)=Ne^{-x}F_N(x)>0\), the real logarithm is well defined
85 for all \(x\ge0\). At infinity,
86 
87 \[
88 \log F_N(x)=x-\log N+O(e^{-c_Nx}),
89 \qquad g_N(x):=\frac{F_N'(x)}{F_N(x)}
90 =1+O(e^{-c_Nx}).
91 \tag{S3}
92 \]
93 
94 At the origin \(g_N(x)=O(x^{N-1})\). Its only Taylor powers are
95 \(x^{Nm-1}\), with \(m\ge1\).
96 
97 \hypertarget{the-branch-and-absolute-interchange-issue}{%
98 \section{The branch and absolute-interchange
99 issue}\label{the-branch-and-absolute-interchange-issue}}
100 
101 The pre-existing genus-zero product is
102 
103 \[
104 d_N(w)=\prod_j(1-w/w_j).
105 \tag{S4}
106 \]
107 
108 For completeness, the series bound \(|d_N(w)|\le\exp(|w|^{1/N})\) gives
109 order at most \(1/N<1\). Hadamard factorization therefore has genus zero
110 and a constant exponential factor, which equals one because
111 \(d_N(0)=1\). The usual zero-count consequence gives
112 \(\sum_j|w_j|^{-\sigma}<\infty\) for every \(\sigma>1/N\). Along the
113 negative real axis (S3) also shows the order is exactly \(1/N\).
114 
115 It is not enough simply to say that no zero lies on the integration
116 contour. We need a uniform angular gap from the branch cut to
117 interchange the Mellin integral with the zero sum. Here is that gap.
118 
119 For \(|\arg(-w)|\le\pi/2\), choose \(x=(-w)^{1/N}\) with
120 \(|\arg x|\le\pi/(2N)\). For every \(j\ne0\),
121 
122 \[
123 \Re x-\Re(\omega^j x)
124 \ge \left[\cos\frac{\pi}{2N}-\cos\frac{3\pi}{2N}\right]|x|
125 =:\gamma_N|x|>0.
126 \tag{S5}
127 \]
128 
129 Indeed, the nearest competing root is separated from the positive
130 direction by at least \(3\pi/(2N)\), while the distinguished root is
131 within \(\pi/(2N)\). Cosine decreases on \([0,\pi]\). Thus, once
132 \((N-1)e^{-\gamma_N|x|}<1\), the distinguished term \(e^x\) in (S1) is
133 larger in modulus than the sum of all competitors, and \(d_N(w)\ne0\).
134 Consequently, sufficiently large zeros avoid the entire closed left
135 half-plane. There are only finitely many remaining zeros in a bounded
136 disk. None lies on the negative ray, since \(d_N(-t)>0\) for \(t\ge0\).
137 It follows that some \(\eta_N>0\) satisfies
138 
139 \[
140 |\operatorname{Arg}w_j|\le\pi-\eta_N\qquad\text{for every }j.
141 \tag{S6}
142 \]
143 
144 With \(c=\sin(\eta_N/2)>0\), this implies
145 
146 \[
147 |t+w_j|\ge c(t+|w_j|),\qquad t\ge0.
148 \tag{S7}
149 \]
150 
151 To check the elementary estimate, write \(w=re^{i\theta}\):
152 \(|t+w|^2=(t+r)^2-4tr\sin^2(\theta/2)\ge(t+r)^2\cos^2(\theta/2)\).
153 
154 \hypertarget{mellin-representation-in-the-original-strip}{%
155 \section{Mellin representation in the original
156 strip}\label{mellin-representation-in-the-original-strip}}
157 
158 Let \(H_N(t)=d_N(-t)=E_{N,1}(t)\), and \(G_N(t)=H_N'(t)/H_N(t)\).
159 Logarithmic differentiation of (S4), locally uniformly off the zeros,
160 gives
161 
162 \[
163 G_N(t)=\sum_j\frac1{t+w_j},\qquad t\ge0.
164 \tag{S8}
165 \]
166 
167 For \(1/N<\sigma<1\), (S7) yields
168 
169 \[
170 \sum_j\int_0^\infty\frac{t^{-\sigma}}{|t+w_j|}\,dt
171 \le\frac{\pi}{c\sin(\pi\sigma)}\sum_j|w_j|^{-\sigma}<\infty.
172 \tag{S9}
173 \]
174 
175 This is the absolute interchange needed for complex \(s\) with
176 \(\Re s=\sigma\). Euler's beta integral and the reflection identity give
177 
178 \[
179 \int_0^\infty\frac{t^{-s}}{t+w}\,dt
180 =\frac{\pi}{\sin(\pi s)}e^{-s\operatorname{Log}w},
181 \quad 0<\Re s<1,\quad |\arg w|<\pi.
182 \tag{S10}
183 \]
184 
185 First prove this for positive \(w\) by substitution and the beta
186 integral, then extend to the slit \(w\)-plane by holomorphic dependence.
187 On compact subsets of that slit plane the denominator estimate supplies
188 the needed local domination.
189 
190 Combining (S8)--(S10), and substituting \(t=x^N\), proves
191 
192 \[
193 \boxed{\mathcal Z_N(s)=\frac{\sin(\pi s)}{\pi}
194 \int_0^\infty x^{-Ns}g_N(x)\,dx,
195 \qquad \frac1N<\Re s<1.}
196 \tag{S11}
197 \]
198 
199 For \(x>0\), \(x^{-Ns}=e^{-Ns\log x}\) uses the real logarithm.
200 Multiplicities are already incorporated in (S4), so no simple-zero
201 assumption entered the proof.
202 
203 \hypertarget{global-continuation-and-the-only-pole}{%
204 \section{Global continuation and the only
205 pole}\label{global-continuation-and-the-only-pole}}
206 
207 One can see directly that \(d_N\) has no zero on \(|w|\le1\):
208 
209 \[
210 |d_N(w)-1|\le\sum_{n\ge1}\frac1{(Nn)!}
211 \le\cosh(1)-1<1.
212 \]
213 
214 The strict inequality also gives a slightly larger zero-free disk by
215 continuity. Thus \(G_N(t)\) has a Taylor series converging uniformly on
216 \([0,1]\),
217 
218 \[
219 G_N(t)=\sum_{k\ge0}a_{N,k}t^k,
220 \qquad g_N(x)=\sum_{m\ge1}N a_{N,m-1}x^{Nm-1}.
221 \tag{S12}
222 \]
223 
224 For any integer \(K\ge0\), define
225 
226 \[
227 \begin{aligned}
228 B_{N,K}(s)={}&
229 \int_0^1x^{-Ns}\left[g_N(x)-\sum_{m=1}^{K}N a_{N,m-1}x^{Nm-1}\right]dx\\
230 &+\sum_{m=1}^{K}\frac{a_{N,m-1}}{m-s}
231  +\int_1^\infty x^{-Ns}\bigl[g_N(x)-1\bigr]dx
232  +\frac1{Ns-1}.
233 \end{aligned}
234 \tag{S13}
235 \]
236 
237 The first integral is holomorphic for \(\Re s<K+1\). This includes local
238 uniform domination, since the Taylor remainder is \(O(x^{N(K+1)-1})\).
239 The last integral is entire in \(s\), because (S3) gives exponential
240 decay against every power of \(x\), uniformly on compact \(s\)-sets,
241 including after any number of \(s\)-derivatives. The rational terms
242 exhibit the only possible poles: positive integers \(1,\ldots,K\) and
243 \(1/N\).
244 
245 On their overlapping domains the functions (S13) agree: adding the next
246 Taylor term subtracts its integral and adds precisely the same rational
247 function. They also agree with (S11) in its original strip. Therefore
248 
249 \[
250 \mathcal Z_N(s)=\frac{\sin(\pi s)}{\pi}B_{N,K}(s)
251 \tag{S14}
252 \]
253 
254 provides compatible meromorphic continuations covering the whole plane
255 as \(K\to\infty\). At each positive integer \(m\), the apparent simple
256 pole \(a_{N,m-1}/(m-s)\) cancels with the simple zero of
257 \(\sin(\pi s)\). The term \(1/(Ns-1)\) has residue \(1/N\), and its
258 multiplier at \(s=1/N\) is nonzero. This proves the stated unique simple
259 pole and its residue.
260 
261 It also proves that the original series cannot converge absolutely at
262 \(s=1/N\). If it did, its absolute terms would dominate the series for
263 real \(s\downarrow1/N\), except for finitely many zeros of modulus below
264 one. Dominated convergence would then keep that real limit bounded,
265 contradicting the nonzero pole. Absolute convergence at any smaller real
266 part would imply convergence at \(1/N\). The abscissa is consequently
267 exactly \(1/N\).
268 
269 \hypertarget{trace-zeros-rational-values-and-the-origin}{%
270 \section{Trace zeros, rational values, and the
271 origin}\label{trace-zeros-rational-values-and-the-origin}}
272 
273 For \(\Re s<1\) we may use \(K=0\):
274 
275 \[
276 B_{N,0}(s)=\int_0^1x^{-Ns}g_N(x)\,dx
277 +\int_1^\infty x^{-Ns}(g_N(x)-1)\,dx+\frac1{Ns-1}.
278 \tag{S15}
279 \]
280 
281 This expression is holomorphic at every nonpositive integer. Hence (S14)
282 proves \(\mathcal Z_N(-m)=0\). These statements do not assert that every
283 such zero is simple.
284 
285 At a positive integer \(m\), taking the removable limit in (S14) gives
286 
287 \[
288 \mathcal Z_N(m)=(-1)^{m+1}a_{N,m-1}.
289 \tag{S16}
290 \]
291 
292 If \(c_n=(-1)^n/(Nn)!\), the previous logarithmic-product law therefore
293 remains
294 
295 \[
296 b_m:=\mathcal Z_N(m)
297 =-m c_m-\sum_{j=1}^{m-1}c_jb_{m-j}.
298 \tag{S17}
299 \]
300 
301 In particular all these values are rational. This is the finite
302 coefficient dictionary, independent of numerical zero location.
303 
304 At \(s=0\), the elementary antiderivatives in (S15) give
305 
306 \[
307 \begin{aligned}
308 B_{N,0}(0)
309 &=\log F_N(1)
310  +\lim_{R\to\infty}\bigl[\log F_N(R)-R-\log F_N(1)+1\bigr]-1\\
311 &=-\log N.
312 \end{aligned}
313 \]
314 
315 Since \(\sin(\pi s)/\pi=s+O(s^3)\), (S14) implies \(\mathcal Z_N(0)=0\)
316 and \(\mathcal Z_N'(0)=-\log N\).
317 
318 \hypertarget{classical-benchmarks-and-numerical-examples}{%
319 \section{Classical benchmarks and numerical
320 examples}\label{classical-benchmarks-and-numerical-examples}}
321 
322 The inherited identities
323 
324 \[
325 \mathcal Z_2(s)=\pi^{-2s}\zeta(2s,1/2),\qquad
326 \mathcal Z_4(s)=(\sqrt2\pi)^{-4s}\zeta(4s,1/2)
327 \]
328 
329 now hold as meromorphic identities, by equality in the original
330 half-plane and uniqueness of continuation. Their pole residues are
331 respectively \(1/(2\pi)\) and \(1/(4\sqrt2\pi)\), agreeing with the
332 general theorem. Their zero values at nonpositive integers follow from
333 the Hurwitz formula, and their regularized products are respectively 2
334 and 4.
335 
336 The companion script performs 411 exact rational assertions: independent
337 formal-logarithm and quotient calculations for \(2\le N\le10\) through
338 degree 40; three explicit integer values for each \(N\); and 12
339 Bernoulli/Hurwitz anchors each for \(N=2,4\). It then evaluates (S13)
340 using a truncated local Taylor sum and high-precision exponential-tail
341 quadrature. Sixteen \(N=2,4\) benchmarks sample both sides of the
342 original convergence boundary, negative real values, and complex values.
343 Values for \(N=3,5\) and checks of \(\mathcal Z_N'(0)\) are recorded in
344 the JSON evidence.
345 
346 Those quadrature results are approximations. Agreement at two precisions
347 and with known functions tests normalization and implementation; it does
348 not replace the proof, certify intervals, or settle historical novelty.
349 
350 \hypertarget{a-worked-hurwitz-deformation}{%
351 \section{A worked Hurwitz
352 deformation}\label{a-worked-hurwitz-deformation}}
353 
354 There is a concrete shifted family. For a real shift \(a\ge0\), define
355 
356 \[
357 \mathcal Z_N(s;a)=\sum_j(w_j+a)^{-s},\qquad \Re s>1/N,
358 \tag{T1}
359 \]
360 
361 again with the principal logarithm and multiplicities. A rightward shift
362 preserves the angular gap from the negative ray: for each nonreal \(w\),
363 \(|\arg(w+a)|\) is nonincreasing in \(a\), and positive real zeros
364 remain positive. The zero sequence is unchanged at infinity up to a
365 bounded translation, so the previous absolute convergence argument
366 applies. The normalized determinant is exactly
367 
368 \[
369 \frac{H_N(a+t)}{H_N(a)}
370 =\prod_j\left(1+\frac{t}{w_j+a}\right),\qquad
371 H_N(t)=E_{N,1}(t)>0\quad(t\ge0).
372 \tag{T2}
373 \]
374 
375 This follows by taking the ratio of the two convergent products in (S4),
376 not by introducing an unverified exponential factor. Its logarithmic
377 derivative is \(G_{N,a}(t)=H_N'(a+t)/H_N(a+t)\). The same dominated
378 beta-integral argument gives
379 
380 \[
381 \mathcal Z_N(s;a)=\frac{\sin(\pi s)}{\pi}
382 \int_0^\infty t^{-s}G_{N,a}(t)\,dt,
383 \qquad 1/N<\Re s<1.
384 \tag{T3}
385 \]
386 
387 At infinity, (S2)--(S3), with \(x=(a+t)^{1/N}\), imply
388 
389 \[
390 G_{N,a}(t)=\alpha(t+a)^{\alpha-1}
391 +O\!\left((t+a)^{\alpha-1}e^{-c_N(t+a)^\alpha}\right).
392 \tag{T4}
393 \]
394 
395 For fixed \(a\) and any integer \(K\ge0\), the binomial expansion
396 therefore gives
397 
398 \[
399 G_{N,a}(t)=\sum_{k=0}^{K}\alpha\binom{\alpha-1}{k}a^k t^{\alpha-k-1}
400 +O(t^{\alpha-K-2}).
401 \tag{T5}
402 \]
403 
404 To be explicit about continuation, choose a finite splitting point
405 \(T>2a+1\). Subtract the first \(K+1\) terms of (T5) inside the integral
406 from \(T\) to infinity, and add their integrals back as the rational
407 terms
408 
409 \[
410 \frac{\alpha\binom{\alpha-1}{k}a^k
411  T^{\alpha-k-s}}{s-\alpha+k}.
412 \tag{T6}
413 \]
414 
415 The remainder integral is holomorphic for \(\Re s>\alpha-K-1\). The
416 integral on any compact interval away from zero is entire. At zero,
417 \(G_{N,a}\) is analytic since \(H_N(a)>0\); subtracting any finite
418 Taylor polynomial there extends the other side, with apparent poles only
419 at positive integers. Such Taylor poles cancel against \(\sin(\pi s)\),
420 just as in Section~\ref{global-continuation-and-the-only-pole}. Choosing arbitrarily large subtractions at both
421 ends covers every \(s\in\mathbb C\). The resulting continuations agree
422 on overlaps because all added terms are exactly the integrals of the
423 terms removed.
424 
425 For \(a>0\), the complete pole set is consequently
426 
427 \[
428 \boxed{s=\frac1N-k,\quad k=0,1,2,\ldots,}
429 \]
430 
431 and every pole is simple. Its residue is
432 
433 \[
434 \begin{aligned}
435 \operatorname{Res}_{s=\alpha-k}\mathcal Z_N(s;a)
436 &=\frac{\alpha\sin(\pi(\alpha-k))}{\pi}
437   \binom{\alpha-1}{k}a^k\\
438 &=\boxed{\frac{\sin(\pi/N)}{N\pi}
439           \frac{(1-1/N)_k}{k!}a^k.}
440 \end{aligned}
441 \tag{T7}
442 \]
443 
444 Here \((b)_0=1\) and \((b)_k=b(b+1)\cdots(b+k-1)\). The sign calculation
445 uses both \(\sin(\pi(\alpha-k))=(-1)^k\sin(\pi\alpha)\) and
446 \(\binom{\alpha-1}{k}=(-1)^k(1-\alpha)_k/k!\); their signs cancel. All
447 residues are positive when \(a>0\). At \(a=0\), the secondary
448 coefficients vanish, so the only remaining pole is the unshifted pole at
449 \(1/N\). As before, its nonzero residue makes the absolute convergence
450 abscissa exactly \(1/N\).
451 
452 None of the pole locations \(\alpha-k\) is an integer. Thus the sine
453 factor vanishes at every nonpositive integer without an opposing pole:
454 
455 \[
456 \boxed{\mathcal Z_N(-m;a)=0\quad(m\ge0).}
457 \tag{T8}
458 \]
459 
460 Finally, the one-term subtraction is already convergent at \(s=0\),
461 since \(G_{N,a}(t)-\alpha t^{\alpha-1}=O(t^{\alpha-2})\) and
462 \(\alpha<1\). Let
463 
464 \[
465 B_{N,a}(0)=\int_0^1G_{N,a}(t)dt
466  +\int_1^\infty[G_{N,a}(t)-\alpha t^{\alpha-1}]dt-1.
467 \]
468 
469 Integrating exactly and using \(\log H_N(a+t)-t^\alpha\to-\log N\) gives
470 
471 \[
472 B_{N,a}(0)=-\log N-\log H_N(a).
473 \]
474 
475 The multiplier \(\sin(\pi s)/\pi\) then gives
476 
477 \[
478 \boxed{\mathcal Z_N'(0;a)=-\log[N E_{N,1}(a)],\qquad
479        \exp[-\mathcal Z_N'(0;a)]=N E_{N,1}(a).}
480 \tag{T9}
481 \]
482 
483 For \(N=2\), the product reduces to \(2\cosh\sqrt a\). Four direct
484 quadrature checks at \(a=0,1/4,2,9\) agree with \(-\log(2\cosh\sqrt a)\)
485 at 80-digit working precision. They use a local integral, an exactly
486 integrated algebraic leading tail, and an exponentially decaying
487 remainder; the quadrature does not sample numerical zero locations.
488 These are four additional numerical benchmarks, with the original 411
489 exact assertions unchanged. Results are in
490 \path{evidence/spectral_continuation_checks.json} under
491 \path{shifted_N2_product_benchmarks}.
492 
493 This is a classical Hurwitz-type spectral deformation, with a
494 packet-specific explicit regularized determinant. It is useful because a
495 single shift reveals an entire pole ladder that collapses at the
496 unshifted boundary. No historical-priority claim is made for the
497 deformation or determinant identity.
498 
499 \hypertarget{source-comparison-and-bounded-search}{%
500 \section{Source comparison and bounded
501 search}\label{source-comparison-and-bounded-search}}
502 
503 The continuation proof uses the finite-order Hadamard product and the beta
504 and reflection identities. DLMF~\cite[\S\S5.5, 5.12 and 10.46]{DLMF}
505 provides the special-function identities and identifies the seed as a
506 Mittag--Leffler function. The finite-order factorization theorem is also
507 summarized in Elkies's notes~\cite{Elkies}. The normalization, zero counts,
508 complex branch gap and domination estimates needed here were proved in
509 Sections~\ref{the-exponential-packet-and-its-exact-asymptotic}--\ref{global-continuation-and-the-only-pole}.
510 
511 Mellin subtraction and spectral regularization are established methods.
512 Voros~\cite[\S\S2.1--2.2]{Voros2003}, with its published erratum~\cite{Voros2004},
513 provides a related framework. The present proof checks the hypotheses for
514 this Mittag--Leffler seed directly and does not import a theorem about
515 Riemann zeros. The coefficient approach also has antecedents in the
516 hypergeometric zero spectra of Byrnes, Jiu, Moll and Vignat~\cite{Byrnes2013}.
517 
518 This comparison establishes methodological context, not priority for the
519 particular specialization. The archived research notes preserve the broader
520 search record. A complete historical review of Mittag--Leffler and
521 hyper-Bessel zero-zeta functions remains outside the scope of this volume.
522 
523 \hypertarget{scope-and-remaining-questions}{%
524 \section{Scope and remaining
525 questions}\label{scope-and-remaining-questions}}
526 
527 The local proof establishes global continuation for the unweighted orbit
528 family and its nonnegative real shifts. A functional
529 equation is not supplied or presumed. The only-pole theorem concerns the
530 unshifted case; positive shifts have the full pole ladder in (T7). The
531 pole and trace-zero theorems follow from the exact positive-axis
532 exponential asymptotic, and do not imply an RH-type critical-line
533 property. Residue-weighted sums still require separate multiplicity and
534 convergence analysis. A historical comparison against all Mittag-Leffler
535 and hyper-Bessel zero-zeta literature remains open.
536 
537 The evidence archive distinguishes exact identities, numerical checks and
538 previously corrected implementation failures. None of the open questions
539 listed above is assumed in the continuation proof.

Back to source controls

Back to Special Functions