fricas-CR Introduction.sagews

fricas-CR is an extension of FriCAS

that implements the calculus of differentiable functions over an involutive algebra of symbolic expressions -- functions of complex variables.

To run a customized version of FriCAS in a Sage worksheet

The following Sage/Python command modifies the PATH variable to include $HOME/bin. Sage looks for the fricas executable in this PATH.

os.environ['PATH'] = '%s/bin:%s'%(os.environ['HOME'],os.environ['PATH'])

To typeset the output of FriCAS

A mode in SageMathCloud is just a function that takes as input a string. For example, this function takes whatever the cells input is, executes the code in Fricas, then takes the output and displays it using Markdown:

%sage
from fricas import fricas
def fricas_md(s):
    import re
    t = fricas.eval(s)
    #print t
    t=re.compile(r'\r').sub('',t)
    # mathml overbar
    t=re.compile(r'¯').sub('‾',t,count=0)
    # cleanup FriCAS LaTeX
    t=re.compile(r'\\leqno\(.*\)\n').sub('',t)
    t=re.compile(r'\\sb ').sub('_',t,count=0)
    t=re.compile(r'\\sp ').sub('^',t,count=0)
    # float result type to the right
    t=re.compile(r' *Type: (.*)\n').sub(r'<p style="text-align:right">Type: \1</p>',t,count=0)
    #print t
    md(t, hide=False)

%default_mode fricas_md

)set output mathml off
)set output tex on
)set output algebra off

This version of fricas-CR was compiled on

)version

Value = "FriCAS 2017-08-05 compiled at Fri Dec 22 00:46:50 UTC 2017"

In fricas-CR every symbol has a conjugate that can appear in polynomials and more general expressions.

conjugate(x)

$$\overline x$$

Type: Symbol

conjugate conjugate(x)

$$x$$

Type: Symbol

x+conjugate(x)

$${\overline x}+x$$

Type: Polynomial(Integer)

real(x)==(x+conjugate(x))/2

Type: Void

sin(real x)

Compiling function real with type Variable(x) -> Polynomial(Fraction(Integer))

$$\sin \left( {{{{\overline x}+x} \over 2}} \right)$$

Type: Expression(Integer)

Every symbolic operator has a conjugate but no other assumptions are made.

f:=operator 'f
conjugate f
conjugate f(x)

$$f$$

Type: BasicOperator

$$\overline f$$

Type: BasicOperator

$${\overline f} \left( {x} \right)$$

Type: Expression(Integer)

g:=operator 'g
conjugate g(x)

$$g$$

Type: BasicOperator

$${\overline g} \left( {x} \right)$$

Type: Expression(Integer)

Wirtinger chain rule

fricas-CR implements the Wirtinger derivative

differentiate(f(g x),x)

$${{{{\overline g} _{\verb#" "#} ^{,}} \left( {x} \right)} \ {\overline {{{\overline f} _{\verb#" "#} ^{,}} \left( {{g \left( {x} \right)}} \right)}}}+{{{f _{\verb#" "#} ^{,}} \left( {{g \left( {x} \right)}} \right)} \ {{g _{\verb#" "#} ^{,}} \left( {x} \right)}}$$

Type: Expression(Integer)

differentiate(f(g x),conjugate x)

$${{{f _{\verb#" "#} ^{,}} \left( {{g \left( {x} \right)}} \right)} \ {\overline {{{\overline g} _{\verb#" "#} ^{,}} \left( {x} \right)}}}+{{\overline {{{\overline f} _{\verb#" "#} ^{,}} \left( {{g \left( {x} \right)}} \right)}} \ {\overline {{g _{\verb#" "#} ^{,}} \left( {x} \right)}}}$$

Type: Expression(Integer)

test(differentiate(f(g x),conjugate x)=conjugate differentiate(conjugate f(g x),x))

$$true$$

Type: Boolean

differentiate(f conjugate x,x)

$$\overline {{{\overline f} _{\verb#" "#} ^{,}} \left( {{\overline x}} \right)}$$

Type: Expression(Integer)

differentiate(f conjugate x, conjugate x)

$${f _{\verb#" "#} ^{,}} \left( {{\overline x}} \right)$$

Type: Expression(Integer)

differentiate(f(x,conjugate x),x)

$${\overline {{{\overline f} _{{,2}}} \left( {x, \: {\overline x}} \right)}}+{{f _{{,1}}} \left( {x, \: {\overline x}} \right)}$$

Type: Expression(Integer)

differentiate(f(x,conjugate x),conjugate x)

$${\overline {{{\overline f} _{{,1}}} \left( {x, \: {\overline x}} \right)}}+{{f _{{,2}}} \left( {x, \: {\overline x}} \right)}$$

Type: Expression(Integer)

Conjugation commutes with holomorphic functions and the conjugate derivative is zero due to Cauchy-Riemann.

h:=holomorphic operator 'h
conjugate h(x)

$$h$$

Type: BasicOperator

$$h \left( {{\overline x}} \right)$$

Type: Expression(Integer)

differentiate(h(x),conjugate x)

$$0$$

Type: Expression(Integer)

Many common functions are holomorphic, e.g.

conjugate sin(x)
differentiate(sin x,conjugate x)

$$\sin \left( {{\overline x}} \right)$$

Type: Expression(Integer)

$$0$$

Type: Expression(Integer)

differentiate(sin conjugate x,x)

$$0$$

Type: Expression(Integer)

The following definition for the conjugate of $\sqrt{\ }$

conjugate sqrt(x)

$$1 \over {\sqrt {{1 \over {\overline x}}}}$$

Type: Expression(Integer)

makes $\sqrt{\ }$ a holomorphic function.

differentiate(sqrt x,conjugate x)

$$0$$

Type: Expression(Integer)

conjugate(sqrt(-1))

$$1 \over {\sqrt {-1}}$$

Type: Expression(Integer)

retract %

$$-{\sqrt {-1}}$$

Type: AlgebraicNumber

cin:=conjugate operator(operator('sin))$Expression Integer

$$\overline \sin$$

Type: BasicOperator

cin(x)

$${\overline \sin} \left( {x} \right)$$

Type: Expression(Integer)

conjugate %

$$\sin \left( {x} \right)$$

Type: Expression(Integer)

conjugate(sin(x))

$$\sin \left( {{\overline x}} \right)$$

Type: Expression(Integer)

sin(1- %i)

$$\sin \left( {{1 -i}} \right)$$

Type: Expression(Complex(Integer))

conjugate %

$$\sin \left( {{1+i}} \right)$$

Type: Expression(Complex(Integer))

simplify %

$$\sin \left( {{1+i}} \right)$$

Type: Expression(Complex(Integer))

expand %

$$\sin \left( {{1+i}} \right)$$

Type: Expression(Complex(Integer))

simplify %

$$\sin \left( {{1+i}} \right)$$

Type: Expression(Complex(Integer))