/usr/share/maxima/5.41.0/src/logarc.lisp is in maxima-src 5.41.0-3.
This file is owned by root:root, with mode 0o644.
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;;; The data in this file contains enhancments. ;;;;;
;;; ;;;;;
;;; Copyright (c) 1984,1987 by William Schelter,University of Texas ;;;;;
;;; All rights reserved ;;;;;
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;;; (c) Copyright 1982 Massachusetts Institute of Technology ;;;
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(in-package :maxima)
(macsyma-module logarc)
;;; Logarc and Halfangles
(defmfun $logarc (exp)
(cond ((atom exp) exp)
((arcp (caar exp)) (logarc (caar exp) ($logarc (cadr exp))))
((eq (caar exp) '$atan2)
(logarc '%atan2 (list ($logarc (second exp)) ($logarc (third exp)))))
(t (recur-apply #'$logarc exp))))
(defmfun logarc (f x)
;; Gives the logarithmic form of arc trig and hyperbolic functions
(cond ((eq f '%acos)
;; -%i * log(x + %i*sqrt(1-x^2))
(mul -1 '$%i (take '(%log) (add x (mul '$%i (root (sub 1 (power x 2)) 2))))))
((eq f '%asin)
;; -%i * log(sqrt(1-x^2)+%i*x)
(mul -1 '$%i (take '(%log) (add (mul '$%i x) (root (sub 1 (power x 2)) 2)))))
((eq f '%atan)
;; (log(1 + %i*x) - log(1 - %i*x)) /(2 %i)
(div (sub (take '(%log) (add 1 (mul '$%i x))) (take '(%log) (sub 1 (mul '$%i x))))
(mul 2 '$%i)))
((eq f '%atan2)
;; atan2(y,x) = -%i*log((x + %i*y)/sqrt(x^2+y^2))
(destructuring-bind (y x)
x
(mul -1 '$%i
(take '(%log) (div (add x (mul '$%i y))
(root (add (mul x x) (mul y y)) 2))))))
((eq f '%asinh)
;; log(sqrt(x^2+1)+x)
(take '(%log) (add x (root (add 1 (power x 2)) 2))))
((eq f '%acosh)
;; log(x+sqrt(x-1)*sqrt(x+1))
(take '(%log) (add x (mul (root (add x -1) 2) (root (add x 1) 2)))))
((eq f '%atanh)
;; (log(x+1)-log(1-x))/2
(div (sub (take '(%log) (add 1 x)) (take '(%log) (sub 1 x))) 2))
((member f '(%asec %acsc %acot %asech %acsch %acoth) :test #'eq)
;; asec(x) = acos(1/x), and etc.
(logarc (zl-get (zl-get (get f '$inverse) 'recip) '$inverse) (inv x)))
(t (merror "LOGARC: unrecognized argument: ~M" f))))
(defmfun halfangle (f a)
(and (mtimesp a)
(ratnump (cadr a))
(equal (caddr (cadr a)) 2)
(halfangleaux f (mul 2 a))))
(defun halfangleaux (f a) ;; f=function; a=twice argument
(let ((sw (member f '(%cos %cot %coth %cosh) :test #'eq)))
(cond ((member f '(%sin %cos) :test #'eq)
(mul (halfangleaux-factor f a)
(power (div (add 1 (porm sw (take '(%cos) a))) 2) 1//2)))
((member f '(%tan %cot) :test #'eq)
(div (add 1 (porm sw (take '(%cos) a))) (take '(%sin) a)))
((member f '(%sinh %cosh) :test #'eq)
(mul (halfangleaux-factor f a)
(power (div (add (take '(%cosh) a) (porm sw 1)) 2) 1//2)))
((member f '(%tanh %coth) :test #'eq)
(div (add (take '(%cosh) a) (porm sw 1)) (take '(%sinh) a)))
((member f '(%sec %csc %sech %csch) :test #'eq)
(inv (halfangleaux (get f 'recip) a))))))
(defun halfangleaux-factor (f a)
(cond
((member f '(%sin %cos))
(let ((arg (div (if (eq f '%sin)
($realpart a)
(add ($realpart a) '$%pi))
(mul 2 '$%pi))))
(mul
(power -1 (simplify (list '($floor) arg)))
(sub 1
(mul
(add 1
(power -1 (add (simplify (list '($floor) arg))
(simplify (list '($floor) (mul -1 arg))))))
(simplify (list '($unit_step) (mul -1 ($imagpart a)))))))))
((member f '(%sinh %cosh))
(let ((arg (div (add ($imagpart a) '$%pi) (mul 2 '$%pi)))
(fac (if (eq f '%sinh)
(div (power (power a 2) (div 1 2)) a)
1)))
(mul fac
(power -1 (simplify (list '($floor) arg)))
(sub 1
(mul
(add 1
(power -1 (add (simplify (list '($floor) arg))
(simplify (list '($floor) (mul -1 arg))))))
(simplify (list '($unit_step) ($realpart a))))))))
(t 1)))
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