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• functions
  • Add _eval_expand_trig method for tanh. (#21380 by @oostben and @smichr)

This will be added to https://github.com/sympy/sympy/wiki/Release-Notes-for-1.9.

**References to other Issues or PRs**
Fixes #21365

**Brief description of what is fixed or changed**
- Implemented _eval_expand_trig  method for tanh funciton. expand_trig now works to expand tanh in a few different ways.
- tanh(coeff*x) will be expanded to coeff\*tanh(x).
- tanh now detects double and half angle formulas, so if coeff is 2 or .5, it will be expanded according to the following formulas:
    - tanh Half Angle Formula: **tanh(.5\*x) = (cosh(x)-1)/sinh(x)**
    - tanh Double Angle Formula: **tanh(2x) = 2\*tanh(x)/(1+tanh^2(x))**
- _eval_expand_trig also expands tanh(x+y) using the sum of arguments identity.
    - tanh Sum of Arguments Identity: **tanh(x+y) = (tanh(x)+tanh(y))/(1+tanh(x)\*tanh(y))**

**Release Notes**
<!-- BEGIN RELEASE NOTES -->
* functions
  * Add _eval_expand_trig method for tanh. 
<!-- END RELEASE NOTES -->

Example:
```
>>> from sympy import *
>>> x = Symbol('x')
>>> y = Symbol('y')

>>> expand_trig(tanh(x + y))
(tanh(x) + tanh(y))/(tanh(x)*tanh(y) + 1)

>>> expand_trig(tanh(2*x))
2*tanh(x)/(tanh(x)**2 + 1)

>>> expand_trig(tanh(3*x))
3*tanh(x)

>>> expand_trig(tanh(y*x))
tanh(x*y)

>>> expand_trig(tanh((1/2)*x))
(cosh(x) - 1)/sinh(x)
```

Update

The release notes on the wiki have been updated.

I was able to generalize the tanh(coeff*x) case using this formula:

I also removed the pointless multiplication pointed out in the review.

Does this also work with n-ary addition?
(e.g. expand_trig(tan(x+y+z)) -> (-tan(x)*tan(y)*tan(z) + tan(x) + tan(y) + tan(z))/(-tan(x)*tan(y) - tan(x)*tan(z) - tan(y)*tan(z) + 1))

Yes it does work with n-ary addition.

>>> from sympy import *
>>> w = Symbol('w')
>>> x = Symbol('x')
>>> y = Symbol('y')
>>> z = Symbol('z')
>>> expand_trig(tanh(x+y+z))
(tanh(x)*tanh(y)*tanh(z) + tanh(x) + tanh(y) + tanh(z))/(tanh(x)*tanh(y) + tanh(x)*tanh(z) + tanh(y)*tanh(z) + 1)
>>> expand_trig(tanh(x+x+x))
(tanh(x)**3 + 3*tanh(x))/(3*tanh(x)**2 + 1)
>>> expand_trig(tanh(3*x))
(tanh(x)**3 + 3*tanh(x))/(3*tanh(x)**2 + 1)
>>> expand_trig(tanh(x+y))
(tanh(x) + tanh(y))/(tanh(x)*tanh(y) + 1)
>>> expand_trig(tanh(x-y))
(tanh(x) - tanh(y))/(-tanh(x)*tanh(y) + 1)
>>> expand_trig(tanh((1/2)*x))
(cosh(x) - 1)/sinh(x)
>>> expand_trig(tanh(4*x))
(4*tanh(x)**3 + 4*tanh(x))/(tanh(x)**4 + 6*tanh(x)**2 + 1)
>>> expand_trig(tanh(w+x+y+z))
(tanh(w)*tanh(x)*tanh(y) + tanh(w)*tanh(x)*tanh(z) + tanh(w)*tanh(y)*tanh(z) + tanh(w) + tanh(x)*tanh(y)*tanh(z) + tanh(x) + tanh(y) + tanh(z))/(tanh(w)*tanh(x)*tanh(y)*tanh(z) + tanh(w)*tanh(x) + tanh(w)*tanh(y) + tanh(w)*tanh(z) + tanh(x)*tanh(y) + tanh(x)*tanh(z) + tanh(y)*tanh(z) + 1)

Half angle formula wasn't used for expanding tan.
I'm not sure if this can be added later, but at least I think that it's better to align the behaviors

This may need some tests at test_trigonometric.py

OP could be updated to reflect that half angle recognition is not included.

I wonder if expand_trig (or user?) should just be using the conversions from hyperbolic to regular trig functions ala fu.hyper_as_trig. I also see that the tan._eval_trig has real and imaginary handling -- is this not needed for hyperbolic tanh?

The unconditional else will cause integer arguments to expand in terms of tanh(1) -- I'm not sure that is helpful. Someone can replace int args with int*Symbol('1') and expand to see that result if they want.

tanh(5), tanh(5*Symbol('1')) -> (tanh(1)**5 + 5*tanh(1) + 10*tanh(1)**3)/(1 + 5*tanh(1)**4 + 10*tanh(1)**2)

is this not needed for hyperbolic tanh?

The imaginary unit I appears in the formulas for tan because of Euler's formula

    cos(x) + I*sin(x) = exp(I*x).

It could be avoided by elementary plane trigonometric arguments but it simplifies the mathematics.

I is not needed for computations with tanh since it does not appear in 'hyperbolic Euler's formula'

    cosh(x) + sinh(x) = exp(x).

Is there anything else that needs to be done for this PR to be merged?

Should I add some tests for this anywhere?

Where would I note that half angle recognition is not included @smichr?

Is there anything else that needs to be done for this PR to be merged?

please address this comment

Originally we had this with the unconditional else:

expand_trig(tanh(5)) -> (tanh(1)**5 + 5*tanh(1) + 10*tanh(1)**3)/(1 + 5*tanh(1)**4 + 10*tanh(1)**2)
expand_trig(tanh(5*Symbol('1'))) -> (tanh(1)**5 + 10*tanh(1)**3 + 5*tanh(1))/(5*tanh(1)**4 + 10*tanh(1)**2 + 1)

After making the unconditional else into:
elif arg.is_Mul:

I get:
expand_trig(tanh(5)) -> tanh(5)
expand_trig(tanh(5*Symbol('1'))) -> (tanh(1)**5 + 10*tanh(1)**3 + 5*tanh(1))/(5*tanh(1)**4 + 10*tanh(1)**2 + 1)

Also I wanted to note that the reason, I made that else unconditional was because that is how it's done in tan's _eval_expand_trig method:

expand_trig(tan(5)) -> (-10*tan(1)**3 + 5*tan(1) + tan(1)**5)/(-10*tan(1)**2 + 1 + 5*tan(1)**4)

However, I agree that it doesn't make sense to treat all expansions as multiplications if they are not additions. Might be something to change in tan's _eval_expand_trig to keep the behavior standard.

Is this the type of fix you were looking for?

Is this the type of fix you were looking for?

yes -- I am not an expert in trig expression wrangling, but the expansion doesn't seem profitable (and the user can use the symbolic number to expand a certain factor of the numeric arg if they desire, e.g. tan(4) = tan(2*Symbol('2')))

I think it would be ok to make this change in the regular tan expansion, too.

ok to make this change in the regular tan expansion, too.

Are you willing to do that? If so, I'll wait until that change is present before starting tests.

@smichr I added the change to all the trig functions in #21440 because they all share the same behavior.

This is in, thanks. Will watch for #21440 to finish up, too.