simplify trigonometric and hyperbolic functions with argument I*something

[behackl]

The code here currently implements the simplifications sin(I*something) --> I*sinh(something) and cos(I*something) --> cosh(something).

Performance probably is a bit worse (a call to real_part is needed, which is python/cython).

This should only be considered mergeable after the following TODOs are done:

• Simplifications for all trigonometric functions trig function with argument of form I*x #155
• Simplifications for all hyperbolic functions hyperbolic function with argument of form I*x #159
• Check for performance issues

[rwst]

This is pretty optimal, hard to see how to make it faster.

[behackl]

You are right; I've overestimated the cost of the additional real_part call.

The following behavior seems inconsistent to me:

sage: coth(0)
Infinity
sage: cot(0)
Traceback (most recent call last):
...
RuntimeError: cotan_eval(): cot(0) encountered

We should probably stop raising errors at poles and simply return UnsignedInfinity instead.

[rwst]

**********************************************************************
File "src/sage/symbolic/pynac.pyx", line 1426, in sage.symbolic.pynac.py_sin
Failed example:
    sin(QQbar(I))
Exception raised:
    Traceback (most recent call last):
      File "/home/ralf/sage/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 496, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/home/ralf/sage/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 858, in compile_and_execute
        exec(compiled, globs)
      File "<doctest sage.symbolic.pynac.py_sin[3]>", line 1, in <module>
        sin(QQbar(I))
      File "sage/symbolic/function.pyx", line 841, in sage.symbolic.function.GinacFunction.__call__ (build/cythonized/sage/symbolic/function.cpp:9757)
        res = super(GinacFunction, self).__call__(*args, **kwds)
      File "sage/symbolic/function.pyx", line 988, in sage.symbolic.function.BuiltinFunction.__call__ (build/cythonized/sage/symbolic/function.cpp:11320)
        res = super(BuiltinFunction, self).__call__(
      File "sage/symbolic/function.pyx", line 505, in sage.symbolic.function.Function.__call__ (build/cythonized/sage/symbolic/function.cpp:7132)
        res = g_function_eval1(self._serial,
      File "sage/structure/element.pyx", line 1798, in sage.structure.element.RingElement.__mul__ (build/cythonized/sage/structure/element.c:16400)
        return coercion_model.bin_op(left, right, mul)
      File "sage/structure/coerce.pyx", line 1069, in sage.structure.coerce.CoercionModel_cache_maps.bin_op (/home/ralf/sage/src/build/cythonized/sage/structure/coerce.c:9736)
        raise TypeError(arith_error_message(x,y,op))
    TypeError: unsupported operand parent(s) for '*': 'Algebraic Field' and 'Number Field in I with defining polynomial x^2 + 1'
**********************************************************************
File "src/sage/symbolic/pynac.pyx", line 1448, in sage.symbolic.pynac.py_cos
Failed example:
    cos(QQbar(I))
Exception raised:
    Traceback (most recent call last):
      File "/home/ralf/sage/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 496, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/home/ralf/sage/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 858, in compile_and_execute
        exec(compiled, globs)
      File "<doctest sage.symbolic.pynac.py_cos[3]>", line 1, in <module>
        cos(QQbar(I))
      File "sage/symbolic/function.pyx", line 841, in sage.symbolic.function.GinacFunction.__call__ (build/cythonized/sage/symbolic/function.cpp:9757)
        res = super(GinacFunction, self).__call__(*args, **kwds)
      File "sage/symbolic/function.pyx", line 988, in sage.symbolic.function.BuiltinFunction.__call__ (build/cythonized/sage/symbolic/function.cpp:11320)
        res = super(BuiltinFunction, self).__call__(
      File "sage/symbolic/function.pyx", line 505, in sage.symbolic.function.Function.__call__ (build/cythonized/sage/symbolic/function.cpp:7132)
        res = g_function_eval1(self._serial,
      File "sage/structure/element.pyx", line 1798, in sage.structure.element.RingElement.__mul__ (build/cythonized/sage/structure/element.c:16400)
        return coercion_model.bin_op(left, right, mul)
      File "sage/structure/coerce.pyx", line 1069, in sage.structure.coerce.CoercionModel_cache_maps.bin_op (/home/ralf/sage/src/build/cythonized/sage/structure/coerce.c:9736)
        raise TypeError(arith_error_message(x,y,op))
    TypeError: unsupported operand parent(s) for '*': 'Algebraic Field' and 'Number Field in I with defining polynomial x^2 + 1'
**********************************************************************

[behackl]

I'm a bit irritated where the Number Field in I with defining polynomial x^2 + 1 actually comes from; pynac's I should be from SR, and QQbar(I) is from Algebraic Field.

Nevertheless, related to #164?

[behackl]

Nevermind, just found out that this is in fact pynac's I.

[rwst]

Yes, I is special 8) Yes #164 is related but not the cause. If you set catch throw in sage -gdb you will find it's a coercion problem.

[rwst]

I'm very sorry that I haven't looked at this for so long. I have now fixed the conflict, I get a funny fail:

File "src/sage/functions/log.py", line 83, in sage.functions.log.Function_exp.__init__
Failed example:
    exp(log(2) + 2*k*pi*I)
Expected:
    2
Got:
    cosh(-log(2)) - sinh(-log(2))

[rwst]

I'll implement sinh/cosh(log(x)) then merge the PR on top of it.