A resonance whose sign is symbolic is written in j = -a/b in place of i

[Rafael-SOWNet]

Part of #718. Reopens #1852's decline with an answer.

e^(x/n) sin(x sqrt(-1/n^2)) is resonant: 1/n + i sqrt(-1/n^2) is zero when n > 0, and 1/n - i sqrt(-1/n^2) is zero when n < 0. #1852 declined it because neither sign could be chosen. The rule for a polynomial times an exponential times a trigonometric function now writes Euler's identity in j = -a/b instead of i when it cannot show that either term vanishes. Here a is the exponential's rate and b is the frequency. Where the case is resonant at all, j^2 = -1 holds whichever sign n has, and a + j b is zero by construction, so one answer holds on both sides. If b evaluates to zero, the case is still declined.

A logarithm's substitution produces the same integrand, so Rubi's 4.7.5 rows such as x^m sin(a + ln(k x^n) sqrt(-(1 + m)^2/n^2))^p are answered too.

Sizes, measured on master fb527ae, on 2.5.0 and on this branch:

Integrand	2.5.0	master fb527ae	branch
exp(x/n)*sin(x*sqrt((-1)/n^2))	149 chars, no value anywhere	unevaluated	110 chars
exp(x/n)*sin(a + x*sqrt((-1)/n^2))	157 chars, no value anywhere	unevaluated	254 chars
sin(a + ln(k*x^n)*sqrt((-1)/n^2))	unevaluated	unevaluated	339 chars
x^m*sin(a + ln(k*x^n)*sqrt(-(1 + m)^2/n^2))	unevaluated	unevaluated	465 chars
x^m*cos(a + ln(k*x^n)*sqrt(-1/4*(1 + m)^2/n^2))^2	unevaluated	unevaluated	1,027 chars
x^2*sin(a + ln(k*x^n)*sqrt((-1)/n^2))^3	unevaluated	unevaluated	1,873 chars
x^m*sin(a + ln(k*x^2)*sqrt(-1/36*(1 + m)^2))^3	unevaluated	unevaluated	2,049 chars

The test differentiates each of these back at n = 3 and at n = -3.

Rubi corpus, compared with master fb527ae:

• 4.7.5 and all of 2.3 (684 problems): 592 → 617 solved, with 0 wrong on either side.
• The 25 that moved all went from unevaluated to solved, and each was re-run on its own.
• Family 4 sample: 999 → 1001.
• Every other family is unchanged.
• 4.7.5 alone: 121 → 146 of 160. The 14 that remain are all secants and cosecants, a separate seam: 13 unevaluated and 1 timeout.

The allocation gate passed. The suite passed: 15,293 passed, 0 failed.

The #1852 entry in BREAKING-CHANGES.md said the resonance "is declined now". It is amended to the answer, and so is the 4.7.5 entry's sentence that left a symbolic b alone.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura