Finite certified power-series evaluation for Fresnel, erf, and Gauss functions

Inputs are a target label $T\in \{C,S,\mathrm{erf},{}_2{F}_1\}$, an exact dyadic complex argument $z$, and, for the Gauss-series target, exact dyadic complex parameters $a,b,c$. Also input positive exact rational absolute tolerance ${\epsilon }_a$, optional positive exact rational relative tolerance ${\epsilon }_r$, a fixed exponent range $(e_{\min },e_{\max })$, and finite positive integer caps $S_{\max },N_{\max },p_{\max },W$, with $p_{\max }\ge 2$. The stage cap $S_{\max }$ bounds retries, $N_{\max }$ bounds series terms, $p_{\max }$ bounds dyadic precision, and $W$ bounds this procedure's exact interval operations, term constructions, and elementary-DAG node operations. Every nested elementary-range call shares this counter and is interrupted with inconclusive before its next operation would exceed the remaining budget. At stage $q$, use precision $p=\max (2,\min (q,p_{\max }))$ and truncation index $q$ for every elementary-range subcall.

Reject an invalid target, nonpositive tolerance, invalid cap, invalid exponent range $e_{\min }>e_{\max }$, or a Gauss input with $\mathrm{\mid }z{\mathrm{\mid }}^2\ge 1$ or $c\in \{0,-1,-2,\dots \}$ by returning domain. The Fresnel and error-function branches accept every finite complex $z$. Outputs are a finite interval or rectangle with status success, or a status inconclusive, domain, or range as in Certified special-function enclosure result.

All rational and interval arithmetic uses Outward-rounded interval, rectangle and ball operations. Embed real coefficient intervals $X$ as $X+i[0,0]$ in complex arithmetic; typed elementary DAG nodes never use an implicit real-to-complex conversion. If a required intermediate operation or output endpoint overflows its selected format, return range. If an elementary-range subcall returns a non-finite status, skip the current candidate; if no later stage succeeds, return inconclusive. The same candidate-skip rule applies to every other non-finite arithmetic status, including split-domain or entire when a rounded divisor contains zero; never use such a status as a finite interval. Charge each exact arithmetic operation, coefficient term, rounding, and elementary-range call against $W$; before an operation that would exceed $W$, return inconclusive.

At each stage, form $P_q$, a certified dyadic interval containing $\pi$, only when $T\in \{C,S,\mathrm{erf}\}$. For $r\in \{1\mathrm{/}5,1\mathrm{/}239\}$, form $A_q(r)={\sum }_{j=0}^{q-1}(-1{)}^j{r}^{2j+1}\mathrm{/}(2j+1)$ and $E_q(r)=r^{2q+1}\mathrm{/}((2q+1)(1-r^2))$ by exact rational arithmetic. The arctangent series and geometric-tail argument in Range-reduced exponential, logarithm and arctangent series have certified tails gives $\arctan r\in [A_q(r)-E_q(r),A_q(r)+E_q(r)]$. Combine these intervals as $16\arctan (1\mathrm{/}5)-4\arctan (1\mathrm{/}239)$, then round the resulting endpoints outward to the selected format at precision $p$. The Machin identity in Trigonometric identities, Machin's formula and reduced Taylor tails proves $\pi \in P_q$. If this conversion is non-finite, skip this stage for the selected Fresnel or erf target. For the Gauss-series target, omit $P_q$ and all square-root calls.

Only when $T=\mathrm{erf}$, call Certified range evaluation for elementary real and complex functions on the one-node real DAG $\sqrt{P_q}$ with the same exponent range and precision $p$, and truncation index $q$. If it returns a finite interval $R_q$, continue only if $\inf R_q>0$; otherwise skip this stage. This call encloses $\sqrt{\pi }$ by Certified real elementary-function evaluations enclose their ranges. The $C,S$ branches use $P_q$ directly and do not call square root. For a complex rectangle $B=X+iY$, write $M(B)=\max (\mathrm{\mid }\inf X\mathrm{\mid },\mathrm{\mid }\sup X\mathrm{\mid })+\max (\mathrm{\mid }\inf Y\mathrm{\mid },\mathrm{\mid }\sup Y\mathrm{\mid })$; this rational quantity bounds the modulus of every point in $B$.

At each stage $q=1,\dots ,\min (S_{\max },N_{\max },p_{\max })$, form the following candidate for the selected target.

• For $C$, sum its defining terms $c_0,\dots ,c_{q-1}$ from Fresnel, error, and Gauss hypergeometric series by interval arithmetic, using $P_q\mathrm{/}2$ for $\pi \mathrm{/}2$. Let $c_q$ be the first omitted term and set $r_C=\frac{(\sup P_q{)}^{2}M(z{)}^4}{4(2q+1)(2q+2)}\mathrm{.}$ If $r_C<1$, enlarge the partial-sum rectangle in both coordinates by $M(c_q)\mathrm{/}(1-r_C)$; otherwise skip this stage.
• For $S$, sum its first $q$ defining terms using $P_q\mathrm{/}2$, let $s_q$ be the first omitted term, and set $r_S=\frac{(\sup P_q{)}^{2}M(z{)}^4}{4(2q+2)(2q+3)}\mathrm{.}$ If $r_S<1$, enlarge the partial-sum rectangle in both coordinates by $M(s_q)\mathrm{/}(1-r_S)$; otherwise skip this stage.
• For erf, use the interval $2\mathrm{/}{R}_q$ from the single square-root call above for $2\mathrm{/}\sqrt{\pi }$, sum the first $q$ defining terms, and let $e_q$ be the first omitted term. Set $r_E=M(z{)}^2\mathrm{/}(q+1)$. If $r_E<1$, enlarge the partial-sum rectangle in both coordinates by $M(e_q)\mathrm{/}(1-r_E)$; otherwise skip this stage.
• For the Gauss series, compute its terms by $h_0=1$ and $h_{k+1}=h_{k}z\frac{(a+k)(b+k)}{(c+k)(k+1)}$ using outward interval operations. The exact $c+k$ is nonzero by the domain check, but its rounded enclosure may contain zero; in that case skip this candidate before division. If $a$ or $b$ is a nonpositive integer, let $L$ be one more than the smaller absolute value of those nonpositive integer parameters. If $L\le q$, sum exactly through $h_{L-1}$ and use zero tail. Otherwise put $A=\mathrm{\mid }\mathrm{\Re }a\mathrm{\mid }+\mathrm{\mid }\mathrm{\Im }a\mathrm{\mid },B=\mathrm{\mid }\mathrm{\Re }b\mathrm{\mid }+\mathrm{\mid }\mathrm{\Im }b\mathrm{\mid },C_0=\mathrm{\mid }\mathrm{\Re }c\mathrm{\mid }+\mathrm{\mid }\mathrm{\Im }c\mathrm{\mid },Z_0=\mathrm{\mid }\mathrm{\Re }z\mathrm{\mid }+\mathrm{\mid }\mathrm{\Im }z\mathrm{\mid },$ If $q\le C_0$, skip this stage before forming a quotient. Otherwise set $r_H=Z_0\frac{(1+A\mathrm{/}q)(1+B\mathrm{/}q)}{1-C_0\mathrm{/}q}\mathrm{.}$ If $r_H<1$, sum $h_0,\dots ,h_{q-1}$, let $h_q$ be the first omitted term, and enlarge the partial-sum rectangle in both coordinates by $M(h_q)\mathrm{/}(1-r_H)$; otherwise skip this stage. The geometric bound in Gauss hypergeometric series geometric tail bound applies because $A,B,C_0,Z_0$ dominate the corresponding complex moduli.

Round each candidate's final endpoints outward to precision $p$. Return success at the first stage where its radius is at most ${\epsilon }_a$. If relative tolerance was requested, set relative_certified to true exactly when zero is excluded and the contract's relative-radius inequality holds; failure of that relative test does not prevent absolute success. If no stage produces a passing candidate, return inconclusive. The finite stage loop and the finite elementary-range calls define the procedure completely.