On Thu, Nov 20, 2014 at 7:52 AM, Bill Page <bill.p...@newsynthesis.org> wrote:
> So here (20) is a simpler expression for derivative of arg:
>
> (16) -> abs(x)==sqrt(x*conjugate(x))
> Compiled code for abs has been cleared.
> Compiled code for arg has been cleared.
> 1 old definition(s) deleted for function or rule abs
> Type:
> Void
> (17) -> arg(x)==log(x/abs(x))/%i
> 1 old definition(s) deleted for function or rule arg
> Type:
> Void
> (18) -> arg %i
> Compiling function abs with type Complex(Integer) -> Expression(
> Complex(Integer))
> Compiling function arg with type Complex(Integer) -> Expression(
> Complex(Integer))
>
> (18) - %i log(%i)
> Type:
> Expression(Complex(Integer))
> (19) -> complexNumeric %
>
> (19) 1.5707963267_948966192
> Type:
> Complex(Float)
> (20) -> D(arg(x),x)
> Compiling function abs with type Variable(x) -> Expression(Integer)
> Compiling function arg with type Variable(x) -> Expression(Complex(
> Integer))
>
> _
> - %ix + %i x
> (20) ------------
> _
> 2xx
> Type:
> Expression(Complex(Integer))
>
>
> In general I am a little uncertain if, how and when to deal with
> simplifications of expressions like abs that can be expressed in terms of
> more fundamental/elementary functions. What do you think?
The identity abs(z) = sqrt(z*conjugate(z)) is just the same problem as
for things like exp(x) = E^x, csc(x) = 1/sin(x), sinh(x) =
(exp(x)-exp(-x)/2, asin(x) = -i*log(i*x+sqrt(1-x^2)), asinh(x) =
log(x+sqrt(1+x^2)), ...
Essentially a lot of functions can be written using simpler functions.
The expression is sometimes simpler using abs() and sometimes simpler
using sqrt(x*conjugate(x)), and that is true for all the other cases
too. So a CAS needs to be able to handle both, and allow the user to
convert one to the other. For example in SymPy, we can do:
In [1]: sinh(x)**2+1
Out[1]:
2
sinh (x) + 1
In [2]: (sinh(x)**2+1).rewrite(exp)
Out[2]:
2
⎛ x -x⎞
⎜ℯ ℯ ⎟
⎜── - ───⎟ + 1
⎝2 2 ⎠
In [3]: _.expand()
Out[3]:
2⋅x -2⋅x
ℯ 1 ℯ
──── + ─ + ─────
4 2 4
In general, my approach is that I try to define the derivative of
abs(x) in the simplest possible way, which seems to be in terms of
abs(x) as well, instead of sqrt(x*conjugate(x)). But the CAS needs to
be able to rewrite it later if needed, because sometimes things can
simplify.
Ondrej
>
> Bill.
>
> On 20 November 2014 09:41, Bill Page <bill.p...@newsynthesis.org> wrote:
>> ...
>>
>> I had a thought. I suppose that all non-analytic (nonholomorphic)
>> functions of interest can be written in terms of conjugate and some analytic
>> functions, e.g.
>>
>> abs(x)=sqrt(x*conjugate(x))
>>
>> so perhaps all we really need is to know how to differentiate conjugate
>> properly?
>>
>> Bill
>
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