Unanswered Questions

This page lists MaplePrimes questions that have not yet received an answer

Hello,

     Can you enter debug in a worksheet vs a proc?   

    I have a for loop that is crashing in the middle with a bad calculated number to provide to a thermoprops function.   Is there a way to single step through a for loop (1500 iterations) to see which line is getting tripped up?  I can not post the worksheet as it's private development.   The question is generic to any "for loop".

   I have also tried "porting" into a code edit region, but still can not enter the debugger mode.   Is proper debugger only available if I turn the "for loop" into a proc?   If yes, what is the availability of packages called right after restart for the worksheet for a proc, or do all the pre-calcs developed in the worksheet and the packages to be variables fed into the proc or packages re-declared insde a new proc ???

  Sorry if this is a trival question.

Thanks in advance,
Bill
 

I solve numerically a DAE system whose independent variable is named t and the dependent variables are d[1](t), ..., d[n](t).
I would like to to 2D or 3D plots of the solutions and color the resulting curve using a function f(...) of the remaining dependent variables.

Here is a simple example.

 

restart

with(plots):

0

(1)

sys := {
   diff(x(t), t) = v(t)
  ,diff(v(t), t) = cos(t)
  ,x(0) = -1
  ,v(0) = 0

  ,px(t) = piecewise(x(t) >=0, 1, -1)
  ,pv(t) = piecewise(v(t) >=0, 1, -1)
}:

sol := dsolve(sys, numeric):

 

odeplot(sol, [t, x(t), v(t)], t=0..4*Pi)
 

 

# I would like to color this space curve depending on the signs of x(t) and y(t)
#
# for instance, f being a "color function"
f := proc(s)
  local a, b:
  if s::numeric then
    a := round(eval(px(t), sol(s))):
    b := round(eval(pv(t), sol(s))):
    return piecewise(a+b = 2, "Green", a = 1, "Red", b = 1, "Blue", "Gold")
  end if:
end proc:

SOL := proc(s)
  if s::numeric then
    eval([t, x(t), v(t)], sol(s))
  end if:
end proc:


# I would like to make something like this to work

plot3d(SOL(s), s=0..4*Pi, colorfunc=f(s)):  #... which generates a void plot


 

# In some sense a continuous version of this

opts := symbol=solidbox, symbolsize=20:
display( seq(pointplot3d({SOL(s)}, opts, color=f(s)), s in [seq](0..6, 0.1)) );

 

 


 

Download coloring.mw


How can I fix (if possible) the syntax in the command 

plot3d(SOL(s), s=0..4*Pi, colorfunc=f(s)):

???

Thanks in advance

 

 

Has someone tried to connect Grasshopper with Maple yet?

Grasshopper is a visual programming environment on Rhino.

https://en.wikipedia.org/wiki/Grasshopper_3D

Hello;

I am facing error in code. Kindly guide me. Thanks in advance.

1D1P.mw


 

Maple 2021.2 gives error when calling int() first time. Second time it returns the integral unevaluated.

Is this a known issue? But it seems possible to trap this error for now,, which is good.
 

interface(version);

`Standard Worksheet Interface, Maple 2021.2, Windows 10, November 23 2021 Build ID 1576349`

Physics:-Version()

`The "Physics Updates" version in the MapleCloud is 1133 and is the same as the version installed in this computer, created 2022, January 20, 23:5 hours Pacific Time.`

restart;

w:=(7*x - 3 + sqrt(x^2 + (x^3*(x - 1)^2)^(1/3) - x) + sqrt(-2*((-x^2 + x + (x^3*(x - 1)^2)^(1/3)/2)*sqrt(x^2 + (x^3*(x - 1)^2)^(1/3) - x) + x^2*(x - 1))/sqrt(x^2 + (x^3*(x - 1)^2)^(1/3) - x)))/(12*x*(x - 1));

(1/12)*(7*x-3+(x^2+(x^3*(x-1)^2)^(1/3)-x)^(1/2)+(-2*((-x^2+x+(1/2)*(x^3*(x-1)^2)^(1/3))*(x^2+(x^3*(x-1)^2)^(1/3)-x)^(1/2)+x^2*(x-1))/(x^2+(x^3*(x-1)^2)^(1/3)-x)^(1/2))^(1/2))/(x*(x-1))

int(w,x)

Error, (in IntegrationTools:-Indefinite:-AlgebraicFunction) invalid argument for sign, lcoeff or tcoeff

int(w,x)

int((1/12)*(7*x-3+(x^2+(x^3*(x-1)^2)^(1/3)-x)^(1/2)+(-2*((-x^2+x+(1/2)*(x^3*(x-1)^2)^(1/3))*(x^2+(x^3*(x-1)^2)^(1/3)-x)^(1/2)+x^2*(x-1))/(x^2+(x^3*(x-1)^2)^(1/3)-x)^(1/2))^(1/2))/(x*(x-1)), x)

restart;
w:=(7*x - 3 + sqrt(x^2 + (x^3*(x - 1)^2)^(1/3) - x) + sqrt(-2*((-x^2 + x + (x^3*(x - 1)^2)^(1/3)/2)*sqrt(x^2 + (x^3*(x - 1)^2)^(1/3) - x) + x^2*(x - 1))/sqrt(x^2 + (x^3*(x - 1)^2)^(1/3) - x)))/(12*x*(x - 1));
try
  int(w,x);
catch:
  print("error happened");
end try;

(1/12)*(7*x-3+(x^2+(x^3*(x-1)^2)^(1/3)-x)^(1/2)+(-2*((-x^2+x+(1/2)*(x^3*(x-1)^2)^(1/3))*(x^2+(x^3*(x-1)^2)^(1/3)-x)^(1/2)+x^2*(x-1))/(x^2+(x^3*(x-1)^2)^(1/3)-x)^(1/2))^(1/2))/(x*(x-1))

"error happened"

 


 

Download int_problem_jan_21_2022.mw

Dear esteem Colleagues,

Please how do I modify the following two files (though similar) to get consistent errors? I am not sure where I made the mistake.

Any modifications would be appreciated.

Thank you all for your time and mentorship. Best regard

Biratu_Mapleprimes.mw

DDE_2_Mapleprime.mw

When I added kernelopts('assertlevel'=2): now Maple gives an error from call to solve. Also PDEtools:-Solve  gives same error. 

If I remove the  kernelopts('assertlevel'=2): it works OK and gives solution.

Is this known issue? I'd like to keep kernelopts('assertlevel'=2): and still use solve. Maple 2021.2 on windows 10

interface(version);

`Standard Worksheet Interface, Maple 2021.2, Windows 10, November 23 2021 Build ID 1576349`

Physics:-Version();

`The "Physics Updates" version in the MapleCloud is 1131 and is the same as the version installed in this computer, created 2022, January 7, 9:5 hours Pacific Time.`

restart;

3

eq:=1/8*2^(1/2)*(-2*x*(w^4*x^11+(4*w^4-6*w^3)*x^10+(10*w^4-22*w^3+17*w^2)*x^9+(16*w^4-50*w^3+62*w^2-30*w)*x^8+(71/2+19*w^4-72*w^3+132*w^2-114*w)*x^7+(140-685/3*w+16*w^4-76*w^3+176*w^2)*x^6+(263-841/3*w+10*w^4-56*w^3+331/2*w^2)*x^5+(292-1385/6*w+105*w^2+4*w^4-30*w^3)*x^4+(821/4+w^4-733/6*w-10*w^3+91/2*w^2)*x^3+(173/2-2*w^3-122/3*w+11*w^2)*x^2+(77/4+3/2*w^2-37/6*w)*x+3/2-1/2*w)*(-1+I*3^(1/2))^(1/2)+2^(1/2)*(I*x*(w^4*x^11+(4*w^4-6*w^3)*x^10+(10*w^4-22*w^3+17*w^2)*x^9+(16*w^4-50*w^3+62*w^2-30*w)*x^8+(71/2+19*w^4-72*w^3+132*w^2-114*w)*x^7+(140-685/3*w+16*w^4-76*w^3+176*w^2)*x^6+(263-841/3*w+10*w^4-56*w^3+331/2*w^2)*x^5+(292-1385/6*w+105*w^2+4*w^4-30*w^3)*x^4+(821/4+w^4-733/6*w-10*w^3+91/2*w^2)*x^3+(173/2-2*w^3-122/3*w+11*w^2)*x^2+(77/4+3/2*w^2-37/6*w)*x+3/2-1/2*w)*3^(1/2)-1/4-3*w^4*x^12+(-12*w^4+10*w^3)*x^11+(-30*w^4+26*w^3-7*w^2)*x^10+(-48*w^4+54*w^3+14*w^2-14*w)*x^9+(63/2-57*w^4+64*w^3+36*w^2-98*w)*x^8+(140-48*w^4+68*w^3+80*w^2-565/3*w)*x^7+(255-769/3*w-30*w^4+48*w^3+127/2*w^2)*x^6+(292-1169/6*w-12*w^4+34*w^3+45*w^2)*x^5+(797/4-661/6*w+7/2*w^2-3*w^4+14*w^3)*x^4+(173/2-80/3*w+6*w^3-w^2)*x^3+(69/4-25/6*w-9/2*w^2)*x^2+(3/2+3/2*w)*x))/(x^2+x+1)^4/x^4;

(1/8)*2^(1/2)*(-2*x*(w^4*x^11+(4*w^4-6*w^3)*x^10+(10*w^4-22*w^3+17*w^2)*x^9+(16*w^4-50*w^3+62*w^2-30*w)*x^8+(71/2+19*w^4-72*w^3+132*w^2-114*w)*x^7+(140-(685/3)*w+16*w^4-76*w^3+176*w^2)*x^6+(263-(841/3)*w+10*w^4-56*w^3+(331/2)*w^2)*x^5+(292-(1385/6)*w+105*w^2+4*w^4-30*w^3)*x^4+(821/4+w^4-(733/6)*w-10*w^3+(91/2)*w^2)*x^3+(173/2-2*w^3-(122/3)*w+11*w^2)*x^2+(77/4+(3/2)*w^2-(37/6)*w)*x+3/2-(1/2)*w)*(-1+I*3^(1/2))^(1/2)+2^(1/2)*(I*x*(w^4*x^11+(4*w^4-6*w^3)*x^10+(10*w^4-22*w^3+17*w^2)*x^9+(16*w^4-50*w^3+62*w^2-30*w)*x^8+(71/2+19*w^4-72*w^3+132*w^2-114*w)*x^7+(140-(685/3)*w+16*w^4-76*w^3+176*w^2)*x^6+(263-(841/3)*w+10*w^4-56*w^3+(331/2)*w^2)*x^5+(292-(1385/6)*w+105*w^2+4*w^4-30*w^3)*x^4+(821/4+w^4-(733/6)*w-10*w^3+(91/2)*w^2)*x^3+(173/2-2*w^3-(122/3)*w+11*w^2)*x^2+(77/4+(3/2)*w^2-(37/6)*w)*x+3/2-(1/2)*w)*3^(1/2)-1/4-3*w^4*x^12+(-12*w^4+10*w^3)*x^11+(-30*w^4+26*w^3-7*w^2)*x^10+(-48*w^4+54*w^3+14*w^2-14*w)*x^9+(63/2-57*w^4+64*w^3+36*w^2-98*w)*x^8+(140-48*w^4+68*w^3+80*w^2-(565/3)*w)*x^7+(255-(769/3)*w-30*w^4+48*w^3+(127/2)*w^2)*x^6+(292-(1169/6)*w-12*w^4+34*w^3+45*w^2)*x^5+(797/4-(661/6)*w+(7/2)*w^2-3*w^4+14*w^3)*x^4+(173/2-(80/3)*w+6*w^3-w^2)*x^3+(69/4-(25/6)*w-(9/2)*w^2)*x^2+(3/2+(3/2)*w)*x))/((x^2+x+1)^4*x^4)

solve(eq = 0,w)

(1/2)*(2*x^2+1)/((x^2+x+1)*x)

PDEtools:-Solve(eq=0,w)

w = (1/2)*(2*x^2+1)/((x^2+x+1)*x)

restart;

interface(warnlevel=4);
kernelopts('assertlevel'=2):

3

eq:=1/8*2^(1/2)*(-2*x*(w^4*x^11+(4*w^4-6*w^3)*x^10+(10*w^4-22*w^3+17*w^2)*x^9+(16*w^4-50*w^3+62*w^2-30*w)*x^8+(71/2+19*w^4-72*w^3+132*w^2-114*w)*x^7+(140-685/3*w+16*w^4-76*w^3+176*w^2)*x^6+(263-841/3*w+10*w^4-56*w^3+331/2*w^2)*x^5+(292-1385/6*w+105*w^2+4*w^4-30*w^3)*x^4+(821/4+w^4-733/6*w-10*w^3+91/2*w^2)*x^3+(173/2-2*w^3-122/3*w+11*w^2)*x^2+(77/4+3/2*w^2-37/6*w)*x+3/2-1/2*w)*(-1+I*3^(1/2))^(1/2)+2^(1/2)*(I*x*(w^4*x^11+(4*w^4-6*w^3)*x^10+(10*w^4-22*w^3+17*w^2)*x^9+(16*w^4-50*w^3+62*w^2-30*w)*x^8+(71/2+19*w^4-72*w^3+132*w^2-114*w)*x^7+(140-685/3*w+16*w^4-76*w^3+176*w^2)*x^6+(263-841/3*w+10*w^4-56*w^3+331/2*w^2)*x^5+(292-1385/6*w+105*w^2+4*w^4-30*w^3)*x^4+(821/4+w^4-733/6*w-10*w^3+91/2*w^2)*x^3+(173/2-2*w^3-122/3*w+11*w^2)*x^2+(77/4+3/2*w^2-37/6*w)*x+3/2-1/2*w)*3^(1/2)-1/4-3*w^4*x^12+(-12*w^4+10*w^3)*x^11+(-30*w^4+26*w^3-7*w^2)*x^10+(-48*w^4+54*w^3+14*w^2-14*w)*x^9+(63/2-57*w^4+64*w^3+36*w^2-98*w)*x^8+(140-48*w^4+68*w^3+80*w^2-565/3*w)*x^7+(255-769/3*w-30*w^4+48*w^3+127/2*w^2)*x^6+(292-1169/6*w-12*w^4+34*w^3+45*w^2)*x^5+(797/4-661/6*w+7/2*w^2-3*w^4+14*w^3)*x^4+(173/2-80/3*w+6*w^3-w^2)*x^3+(69/4-25/6*w-9/2*w^2)*x^2+(3/2+3/2*w)*x))/(x^2+x+1)^4/x^4:

solve(eq = 0,w)

Error, (in Internal:-FactorEasy) assertion failed, Internal:-FactorEasy expects its return value to be of type list(_POWER(POLYNOMIAL, posint)), but computed [_POWER(POLYNOMIAL([0, [z, x, r1, r2, r3], [[[[1], [0, -1]], 0, [[1]]], [[1], 0, [1]], [-3, 0, 1]]], [[[0, [[1]]]]]), 1), _POWER(POLYNOMIAL([0, [z, x, r1, r2, r3], [[[[1], [0, -1]], 0, [[1]]], [[1], 0, [1]], [-3, 0, 1]]], [[[[0, [1]]]]]), FAIL), _POWER(POLYNOMIAL([0, [z, x, r1, r2, r3], [[[[1], [0, -1]], 0, [[1]]], [[1], 0, [1]], [-3, 0, 1]]], [[[[[0, 1]]]]]), FAIL), _POWER(POLYNOMIAL([0, [z, x, r1, r2, r3], [[[[1], [0, -1]], 0, [[1]]], [[1], 0, [1]], [-3, 0, 1]]], [[[[[-1], [0, 1]]], 0, [[[-8], [0, 8]]], 0, [[[-24], [0, 24]]], 0, [[[-32], [0, 32]]], 0, [[[-16], [0, 16 ... , [[[-160], [0, 160]]], [[[-64], [0, 64]]], [[[-16], [0, 16]]]]]), 1)]

PDEtools:-Solve(eq=0,w)

Error, (in Internal:-FactorEasy) assertion failed, Internal:-FactorEasy expects its return value to be of type list(_POWER(POLYNOMIAL, posint)), but computed [_POWER(POLYNOMIAL([0, [z, x, r1, r2, r3], [[[[1], [0, -1]], 0, [[1]]], [[1], 0, [1]], [-3, 0, 1]]], [[[0, [[1]]]]]), 1), _POWER(POLYNOMIAL([0, [z, x, r1, r2, r3], [[[[1], [0, -1]], 0, [[1]]], [[1], 0, [1]], [-3, 0, 1]]], [[[[0, [1]]]]]), FAIL), _POWER(POLYNOMIAL([0, [z, x, r1, r2, r3], [[[[1], [0, -1]], 0, [[1]]], [[1], 0, [1]], [-3, 0, 1]]], [[[[[0, 1]]]]]), FAIL), _POWER(POLYNOMIAL([0, [z, x, r1, r2, r3], [[[[1], [0, -1]], 0, [[1]]], [[1], 0, [1]], [-3, 0, 1]]], [[[[[-1], [0, 1]]], 0, [[[-8], [0, 8]]], 0, [[[-24], [0, 24]]], 0, [[[-32], [0, 32]]], 0, [[[-16], [0, 16 ... , [[[-160], [0, 160]]], [[[-64], [0, 64]]], [[[-16], [0, 16]]]]]), 1)]

 

Download assertion_failed.mw

Contour integration notation

" (∫)[+infinity]^(+infinity)((-x)^(z))/((e)^(x)-1). (ⅆx)/(x)"

 

The limits of integration are intented to indicate a path of integration which begins at + ∞, moves to th e left down the positive real axis, circles the orign once in positive ( counterclockwise) direction, and returns up to the positive real axis to  +∞

-How does this contour look like  in a  graph ?
- the "(ⅆx)/(x)" notation  ?
- calculating this complexe contour integral?

Seems that the concept of the contour integration is similar wit a line integral in real calculus ?

Some more information needed about singularities ( first en second order ..more?)

Nieuwe pagina 1 (hhofstede.nl)

NULL

Download contourintegraal_vraag1.mw

Just wanted to post that I had some data loss because I opened two different workbooks with the same name from different locations.

This could lead to loss of data.

I had code attachments from the "old" workbook implemented in the "new" workbook, while the new code was gone.

Thought always that the round d is reserved for function of two variables x,y , but  that seems to be not the case here ?

restart;

Comparing Different Answers

 

Een antwoord ergens gegeven is

Int(sqrt(x^2+1), x) = (1/2)*x*sqrt(x^2+1)+(1/2)*ln(x+sqrt(x^2+1)) + C                                                             (vb)

 

Mple geeft

 

Int(sqrt(x^2+1),x)=int(sqrt(x^2+1),x)+C[1];

Int((x^2+1)^(1/2), x) = (1/2)*x*(x^2+1)^(1/2)+(1/2)*arcsinh(x)+C[1]

(1)

 

De twee antwoorden lijken nog niet opelkaar !
In het gegeven antwoord staat er een ln en in Maple kan een expressie omgezet worden in ln termen
  

convert(%,ln);

Int((x^2+1)^(1/2), x) = (1/2)*x*(x^2+1)^(1/2)+(1/2)*ln(x+(x^2+1)^(1/2))+C[1]

(2)

(2)  is hetzelfde (vb)

Dezelfde integraal i sook gegeven als

Int(sqrt(x^2+1),x)=((x+sqrt(x^2+1))^2+4*ln(x+sqrt(x^2+1))-(x+sqrt(x^2+1))^(-2))/8+C[2];

Int((x^2+1)^(1/2), x) = (1/8)*(x+(x^2+1)^(1/2))^2+(1/2)*ln(x+(x^2+1)^(1/2))-(1/8)/(x+(x^2+1)^(1/2))^2+C[2]

(3)

Controle

een effectieve manier om twe antwoorden t evergelijken voor hetzelfde probleem is het verschil te berekenen van een vergelijking met de twee integralen

#lhs(%);

#rhs(%%);

 

#diff(lhs(%)-rhs(%)=0,x);

NULL

#diff(f,x);

diff(lhs(%)-rhs(%)=0,x);

(x^2+1)^(1/2)-(1/4)*(x+(x^2+1)^(1/2))*(1+x/(x^2+1)^(1/2))-(1/2)*(1+x/(x^2+1)^(1/2))/(x+(x^2+1)^(1/2))-(1/4)*(1+x/(x^2+1)^(1/2))/(x+(x^2+1)^(1/2))^3 = 0

(4)

simplify(%);

0 = 0

(5)

Strange that  diff(lhs(%)-rhs(%)=0,x);  is translated by 2 d input with round d notation for functions with two variables ?
The two integrals are functions of one variable
diff(f, x)

Download Controleren_dezelfde_antwoord_voo_expressies.mw

system_of_PDE.mw

NULL

NULL

Digits := 30; with(PDEtools); with(plots); Ops1 := numpoints = 100; Ops2 := color = magenta; Ops3 := color = blue; Ops4 := color = "BlueViolet"; Ops5 := axes = boxed, shading = zhue, orientation = [40, 50]; a := 0; b := 1; Tf := .5

axes = boxed, shading = zhue, orientation = [40, 50]

 

.5

(1)

E := 1480

1480

(2)

Ebes := 5990

5990

(3)

n0 := 900000

900000

(4)

ro := 1200

1200

(5)

m := 12.6

12.6

(6)

f := sig(x, t)-Ebes*`ϵij`(x, t)

sig(x, t)-5990*`ϵij`(x, t)

(7)

n := 900000*exp(-(sig(x, t)-E*`ϵij`(x, t))/m)

900000*exp(-0.793650793650793650793650793651e-1*sig(x, t)+117.460317460317460317460317460*`ϵij`(x, t))

(8)

NULL

P := 5

5

(9)

w := 4

4

(10)

k := 7

7

(11)

i := 5

5

(12)

eq1 := diff(sig(x, t), x, x) = ro*(diff(sig(x, t), x, x))/E+ro*(diff(sig(x, t), t)-Ebes*f/(9000000*exp(-(sig(x, t)-E*`ϵij`(x, t))/m)))*(1+f/m)/(9000000*exp(-(sig(x, t)-E*`ϵij`(x, t))/m))

diff(diff(sig(x, t), x), x) = (30/37)*(diff(diff(sig(x, t), x), x))+(1/7500)*(diff(sig(x, t), t)-(599/900000)*(sig(x, t)-5990*`ϵij`(x, t))/exp(-0.793650793650793650793650793651e-1*sig(x, t)+117.460317460317460317460317460*`ϵij`(x, t)))*(1+0.793650793650793650793650793651e-1*sig(x, t)-475.396825396825396825396825397*`ϵij`(x, t))/exp(-0.793650793650793650793650793651e-1*sig(x, t)+117.460317460317460317460317460*`ϵij`(x, t))

(13)

 

eq2 := diff(`ϵij`(x, t), t) = f/(9000000*exp(-(sig(x, t)-E*`ϵij`(x, t))/m))

diff(`ϵij`(x, t), t) = (1/9000000)*(sig(x, t)-5990*`ϵij`(x, t))/exp(-0.793650793650793650793650793651e-1*sig(x, t)+117.460317460317460317460317460*`ϵij`(x, t))

(14)

NULL

sys := {eq1, eq2}

{diff(diff(sig(x, t), x), x) = (30/37)*(diff(diff(sig(x, t), x), x))+(1/7500)*(diff(sig(x, t), t)-(599/900000)*(sig(x, t)-5990*`ϵij`(x, t))/exp(-0.793650793650793650793650793651e-1*sig(x, t)+117.460317460317460317460317460*`ϵij`(x, t)))*(1+0.793650793650793650793650793651e-1*sig(x, t)-475.396825396825396825396825397*`ϵij`(x, t))/exp(-0.793650793650793650793650793651e-1*sig(x, t)+117.460317460317460317460317460*`ϵij`(x, t)), diff(`ϵij`(x, t), t) = (1/9000000)*(sig(x, t)-5990*`ϵij`(x, t))/exp(-0.793650793650793650793650793651e-1*sig(x, t)+117.460317460317460317460317460*`ϵij`(x, t))}

(15)

NULL

IBC1 := {sig(0, t) = P*sin(w*k*i), sig(10, t) = P*sin(w*k*i), sig(x, 0) = 0, sig(x, 1) = 0, `ϵij`(x, 0) = 0}

{sig(0, t) = 5*sin(140), sig(10, t) = 5*sin(140), sig(x, 0) = 0, sig(x, 1) = 0, `ϵij`(x, 0) = 0}

(16)

S := 1/100; Ops := spacestep = S, timestep = S; Sol1 := pdsolve(sys, IBC1, [sig, `ϵij`], numeric, time = t, range = a .. b, Ops)

1/100

 

spacestep = 1/100, timestep = 1/100

 

Error, (in pdsolve/numeric) unable to handle elliptic PDEs

 

``

Download system_of_PDE.mw

I have defined a function II1norm of one variable. The variable has units "microns". It plots perfecting using a range defined in microns, but gives an error when I try to find the root using NextZero. If I just leave off the "microns" in the second argument, NextRoot just reports "FAIL". If I rewite the worksheet without units, then the NextZero executes fine. Why? How to I use units when finding roots?

This is so useful to see geometrical mapping diagram to visualize Complex analysis

Something that also can be made for Maple 

Mapping Diagram for Cauchy Integral Formula – GeoGebra

Using GeoGebra for visualizing complex variable. (google.com)

I highly encourage everyone interested in complex variable to read Tristan Needham „Visual Complex Analysis” and try to solve problems with or without aid of GeoGebra. I hope that in this workshop we will manage to get a feeling of complex functions and as a final point understand how complex integration works. It is a common misconception that complex integration can't be visualized, and using Tristan Needham's ideas we will try to explore this idea. It's a pity that we don't have a lot of time, thus we will skip a lot of important information and construct only some graphs. 

There is so much experimenting with Geogebra software and doing too this in Maple ?

Hello there, 

Would you allow me to ask one question?

Is there any way to get a saturated water vapor pressure value with a temperature outside of the range?

The range here means [273.06 K, 647.096 K]. The pressure value certainly exists (T<273.06 K), but the API only comes up with an error. 

Here is the Maple worksheet where I got into this issue:

restart;

with(ThermophysicalData):

with(CoolProp):        

with(Units[Standard]):

with(ScientificConstants):

T2 := (-40.0 + 273.15) * Unit('K');

233.15*Units:-Unit(K)

(1)

xbb := 1: # 100% steam, saturated.

Pg2 := PropsSI("P", "T", T2, "Q", xbb, "water");

Error, (in ThermophysicalData:-CoolProp:-PropsSI) Temperature to QT_flash [233.15 K] must be in range [273.06 K, 647.096 K] : PropsSI("P","T",233.15,"Q",1,"water")

 

 

Download Q20220111.mw

I must feed the not trivial zeros numbers into this aproximation formula ?

 

Riemann hypothese and staircase of primes

 

restart;

with(NumberTheory)

PrimeCounting(1)

0

(1)

pi(Pi)

2

(2)

PrimeCounting(10000)

1229

(3)

numelems(select(isprime, [seq(1 .. 10000)]))

1229

(4)

The prime counting function is approximated by Li(x) and x/ln(x).

plot([PrimeCounting(x), Li(x), x/ln(x)], x = 1 .. 500, legend = [pi(x), Li(x), x/ln(x)])

The staircase of primes approximated by two functions
Interesting is the video: How i learned to love and fear the Riemann Hypothesis

https://www.quantamagazine.org/how-i-learned-to-love-and-fear-the-riemann-hypothesis-20210104/

NULL

ps:=Array(1..30):
y:=0:
for n from 1 to 30 do
 if is(n,prime)
     then ps[n]:=plot([[n,y],[n,y+1],[n+1,y+1]]):
     y:=y+1;
     else ps[n]:=plot([[n,y],[n+1,y]]):
 end if ;
od;
with(plots):
display({seq(ps[n],n=1..30)}):  

plot([PrimeCounting(x)] ,x = 1 .. 35, legend = [pi(x)]):

plot([PrimeCounting(x), Li(x), x/ln(x)], x = 1 .. 35, legend = [pi(x), Li(x), x/ln(x)])

 

 

 

 

Prime counting function
What found RIEMANN for the prime counting function in relation to the zeta function after he defined the zeta function?

 

He found further a function what follows exactly the shape of the prime counting function

Final discovery v. Riemann.  

- step in the omhoog in de priemtelfunctie = log(p) (zie video)

 

Using the logarithmic primecount function( from Chebyshev) (approximation)
Further  analyse with this Chebyshev approximation formula in relation to the not trivial zero points from Riemann zeta function ( zeros) gives another real function for approximating the primecounting function what uses the non trivial zeros from Riemanns zeta function  in this function:

 

"(not trivial zeros ) u[k ] = "i*w[k]+v[k]   
Number now all nottrivial zeros in the upperhalfplane from down to bottom,  as u[1], u[2], u[3, () .. ()]

"`&varphi;`(x)  := x-ln(2Pi)-1/(2 )ln(1-1/(x^(2))) - (&sum;)2/(|u[k]|) x^(v[k]) cos(w[k] ln(x)-alpha[k])"

NULL

Its only alpha[k] that must be calculated out of the not trivial zeros and i must have a list of  serie of not trivial zeroes from the zeta function. => see Hardy's Z(t) ? from this ..... alpha[k]  can be calculated ?
All not trivial zeros are complex numbers laying on a line ,but  orginating from (0,0) in the complex plane as  vectors to the points    

varphi(x):= x - ln(2*Pi) - 1/2*ln(1 - 1/x^2) - sum(2*x^v[k]*cos(w[k]*ln(x) - alpha[k])/abs(u[k]), k = 1 .. infinity);

x-ln(2*Pi)-(1/2)*ln(1-1/x^2)-(sum(2*x^v[k]*cos(-w[k]*ln(x)+alpha[k])/abs(u[k]), k = 1 .. infinity))

(5)

This formula seems to be correct .
Now how to make a plot ?
Hardy's Z(t) function shows the not trivial zeros in the upperhalfplane of the critical strip of the Riemann zeta function  as zeros in this Z(t) real function : derived from a alternating serie ?

Download priem_staircase_en_riemann_functie.mw

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