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• functions
  • Fix derivative of acosh (#23568 by @eagleoflqj)

This will be added to https://github.com/sympy/sympy/wiki/Release-Notes-for-1.11.

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This justifies #23566
Wolfram Alpha result: https://www.wolframalpha.com/input?i=arccosh%28x%29+derivative
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* functions
  * Fix derivative of acosh
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Update

The release notes on the wiki have been updated.

Change looks good to me but failing test need to be looked into !

The failing tests are because changing the output of diff affects the heurisch algorithm. According to the linked issue, it's not clear if the failing integrals were even correct before.

DLMF (https://dlmf.nist.gov/4.38) gives the derivative as

d/dz arccosh(z) = +- 1/sqrt(z**2 - 1)

where the +- depends on whether z is in the left or right side of the complex plane.

Is that equivalent to 1/(sqrt(z-1)*sqrt(z+1))?

That seems to agree in all quadrants:

In [59]: e1 = sign(re(z))/sqrt(z**2 - 1)

In [60]: e2 = 1/(sqrt(z-1)*sqrt(z+1))

In [61]: e1
Out[61]: 
sign(re(z))
───────────
   ________
  ╱  2     
╲╱  z  - 1 

In [62]: e2
Out[62]: 
         1         
───────────────────
  _______   _______
╲╱ z - 1 ⋅╲╱ z + 1 

In [63]: (e1 - e2).subs(z, 1+I).n()
Out[63]: -0.e-124 + 0.e-124⋅ⅈ

In [64]: (e1 - e2).subs(z, 1-I).n()
Out[64]: -0.e-124 - 0.e-124⋅ⅈ

In [65]: (e1 - e2).subs(z, -1+I).n()
Out[65]: -0.e-134 - 0.e-134⋅ⅈ

In [66]: (e1 - e2).subs(z, -1-I).n()
Out[66]: -0.e-128 + 0.e-125⋅ⅈ

The formulas are perhaps consistent on the imaginary axis if + is taken on the positive axis and - on the negative imaginary axis:

In [83]: e1 = +1/sqrt(z**2 - 1)

In [84]: e2 = 1/(sqrt(z-1)*sqrt(z+1))

In [85]: e1.subs(z, I).n()
Out[85]: -0.707106781186548⋅ⅈ

In [86]: e2.subs(z, I).n()
Out[86]: -0.707106781186548⋅ⅈ

In [87]: e1.subs(z, -I).n()
Out[87]: -0.707106781186548⋅ⅈ

In [88]: e2.subs(z, -I).n()
Out[88]: 0.707106781186548⋅ⅈ

At +-0.5 the formulas seem to contradict:

In [93]: e1 = sign(re(z))/sqrt(z**2 - 1)

In [94]: e2 = 1/(sqrt(z-1)*sqrt(z+1))

In [95]: e1.subs(z, 0.5).n()
Out[95]: -1.15470053837925⋅ⅈ

In [96]: e2.subs(z, 0.5).n()
Out[96]: -1.15470053837925⋅ⅈ

In [97]: e1.subs(z, -0.5).n()
Out[97]: 1.15470053837925⋅ⅈ

In [98]: e2.subs(z, -0.5).n()
Out[98]: -1.15470053837925⋅ⅈ

At least for mpmath's numerical evaluation it seems that the new formula in this PR (e2) agrees with the numerical derivative at +-0.5 and +-I:

In [99]: x = -0.5

In [100]: h = 0.0001

In [101]: N((acosh(x+h) - acosh(x))/h)
Out[101]: -1.15466205349524⋅ⅈ

In [102]: x = I

In [103]: N((acosh(x+h) - acosh(x))/h)
Out[103]: 1.7677669020512e-5 - 0.707106780890414⋅ⅈ

In [104]: x = -I

In [105]: N((acosh(x+h) - acosh(x))/h)
Out[105]: 1.7677669020512e-5 + 0.707106780890414⋅ⅈ

From a casual check this looks good but it would be good to do some serious consistency checking.

Looks like the result for acsch is already the same expression as wolfram rather than the one given in DLMF:

In [6]: acsch(z).diff(z)
Out[6]: 
      -1        
────────────────
        ________
 2     ╱     1  
z ⋅   ╱  1 + ── 
     ╱        2 
   ╲╱        z  

Maybe there should be some tests that randomly substitute values into different functions to check whether the numerical derivative agrees with the symbolic derivative at least under evaluation by mpmath. I expect that a bunch of bugs could be found like this one if all functions were tested.

This justifies #23566

It would be good to have a test for the integral in the issue as well.

it seems that the definition given in wikipedia for complex arguments will agree with the title. It has the proper branch cut (-oo, 1).

Benchmark results from GitHub Actions

Lower numbers are good, higher numbers are bad. A ratio less than 1
means a speed up and greater than 1 means a slowdown. Green lines
beginning with + are slowdowns (the PR is slower then master or
master is slower than the previous release). Red lines beginning
with - are speedups.

Significantly changed benchmark results (PR vs master)

Significantly changed benchmark results (master vs previous release)

       before           after         ratio
     [77f1d79c]       [d0134b30]
     <sympy-1.10.1^0>                 
+         124±2ms          233±5ms     1.89  sum.TimeSum.time_doit

Full benchmark results can be found as artifacts in GitHub Actions
(click on checks at the top of the PR).

Failed (wrong) tests are corrected in other PRs, and integrate(1/sqrt(x**2-1)) == log(x + sqrt(x**2 - 1)) now.
I think this can be merged.