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#2351 Imaginary unit to the power of 4/3

None
closed
nobody
None
5
2012-12-10
2012-02-03
Anonymous
No

(%i1) (%i)^(4/3)
(%o1) 1

WxMaixima
System info
wxWidgets: 2.8.12
Unicode Support: no
Maxima version: 5.26.0
Lisp: GNU Common Lisp (GCL) GCL 2.6.8 (a.k.a. GCL)

Discussion

  • Aleksas
    Aleksas
    2012-02-04

    After replace %i to polarform(%i) or domain:complex; we get correct result:

    (%i1) polarform(%i)^(4/3);
    (%o1) (sqrt(3)*%i)/2-1/2
    (%i2) rectform(%);
    (%o2) (sqrt(3)*%i)/2-1/2

    or
    (%i3) domain:complex;
    (%o3) complex
    (%i4) (%i)^(4/3);
    (%o4) (-1)^(2/3)
    (%i5) rectform(%);
    (%o5) (sqrt(3)*%i)/2-1/2

    Other example: integrate(exp(x^5),x,0,1)
    Wrong:
    (%i6) domain:complex;
    (%o6) complex
    (%i7) integrate(exp(x^5),x,0,1);
    (%o7) (%e^((2*%i*%pi)/5)*(gamma_incomplete(1/5,-1)-gamma(1/5)))/5
    (%i8) float(rectform(%)),expand;
    (%o8) 0.37851290892278-1.164942948399964*%i

    Correct:
    (%i9) assume(k>1)$ declare(k,odd)$
    (%i11) sol:integrate(exp(x^k),x,0,1);
    "Is "(k-1)/k" an "integer"?"n;
    (%o11) (gamma(1/k)/(-1)^(1/k)-gamma_incomplete(1/k,-1)/(-1)^(1/k))/k
    (%i12) subst(k=5,sol);
    (%o12) (gamma(1/5)/(-1)^(1/5)-gamma_incomplete(1/5,-1)/(-1)^(1/5))/5
    (%i13) float(rectform(%)),expand;
    (%o13) 1.1102230246251565*10^-16*%i+1.224893503635311
    (%i14) realpart(%);
    (%o14) 1.224893503635311
    (%i15) quad_qags(exp(x^5), x, 0, 1);
    (%o15) [1.224893503635311,5.5812865751276883*10^-11,21,0]

    Aleksas D

     

  • Anonymous
    2012-02-13

    Thank you kindly for the detailed answer.

    I seem to recall reading that maxima uses complex numbers by default. As a general comment I find it odd that polarform(%i) would not give the same result as just %i.

    However I've learned something. Much appreciated.

     
  • Robert Dodier
    Robert Dodier
    2012-12-10

    • Description has changed:

    Diff:

    --- old
    +++ new
    @@ -1,4 +1,3 @@
    -
     \(%i1\)  \(%i\)^\(4/3\)
     \(%o1\) 1
    
    • status: open --> closed
    • milestone: --> None
     
  • Robert Dodier
    Robert Dodier
    2012-12-10

    Observed behavior is to be expected given the current assumptions about simplification of complex numbers. Marking this report "closed" accordingly.