## Complex Roots

One question which has fascinated me in recent years, was the question of what exactly happens, if we start with a complex number as a base, and if we then raise that base to either a rational exponent, or an irrational exponent.

In the following worksheet, I explored that subject for all to see:

http://dirkmittler.homeip.net/Plot_Complex_Points.pdf

(Note : )

The worksheet above was made using a graphical front-end to the open-source Computer Algebra System named ‘Maxima’. That graphical front-end is named ‘wxMaxima’, and adds as features, to be able to convert textual output from Maxima, into typeset equations, as well as to make certain presentations and plots nicer.

Enjoy,

Dirk

## A New Realization about Common LISP

In This Posting, I explored the potential fact in great depth, that certain implementations of LISP may have Symbols, one field of which holds a value in the sense that a variable should hold a value, consisting of lists and atoms, while another field can hold a property list, also known as a “Symbol Plist”.

Well it has come to my attention, that Common LISP in particular, has some fixed constraints regarding this subject.

When Common LISP evaluates expressions that are lists, the CAR is allowed to be a nested list, which defines a lambda-expression. But otherwise, the CAR needs to be an atom – i.e., a symbol – which has a compiled function, stored in the plist.

We may export function-definitions into the value-field of a symbol, such as into a variable, like so:


(defvar *function-var*
#'(lambda (pos-arg1 pos-arg2)
(list pos-arg1 pos-arg2) ) )



If we do, then we must call or apply this function like so:



(funcall *function-var* 'Foo 'Blatz)

(apply *function-var* '(Foo Blatz) )




If we do not abide by this convention, we get error messages, because the function-definition in the value-field is not searched by Common LISP, when the symbol is itself the CAR of a list, and because when we assign lists as function-definitions, those do not get assigned to the symbol plist.

This observation seems to be corroborated here.

This convention, of Common LISP, accelerates execution greatly, because the interpreter does not need to make complicated arrangements, depending on what type of functions appear as the CAR of a list. The context now specifies that.

Once we are aware that this constraint exists, adapting our value-field-functions to it, is straightforward, because of the built-in function

(symbol-function 'symbol)

Which disembodies the part of the plist, that is supposed to hold functions and not values, in this case, belonging to 'symbol. Hence, the following experiment should succeed on a working Common LISP Compiler:



GCL (GNU Common Lisp)  2.6.12 CLtL1    Oct 28 2014 10:02:30
Modifications of this banner must retain notice of a compatible license
Dedicated to the memory of W. Schelter

Use (help) to get some basic information on how to use GCL.
Temporary directory for compiler files set to /tmp/

>(setf (symbol-function 'my-function)
(lambda (pos-arg1 pos-arg2)
(list pos-arg1 pos-arg2) ) )

(LAMBDA-CLOSURE () () () (POS-ARG1 POS-ARG2) (LIST POS-ARG1 POS-ARG2))

>(my-function 'Foo 'Blatz)

(FOO BLATZ)

>(my-function 'Ergo 'Sum)

(ERGO SUM)

>(quit)




Now, the somewhat unexpected line output from the definition, is due to the (setf) function doing what all variable-setting functions in LISP do: They not only bind the value asked for, to the field, but also return the same value to the calling context, because doing so can make certain LISP code more practical.