MaplePrimes Questions

I have a repetative set of parameter names used in procedures in a package. I settled on making all optional inputs in the format {name::type:=something}. Originally I wanted to use non capitalised names for the optional inputs. But the names clash with Maple commands. I have all sorts of quicky abbreviations like prnt etc. These are both messy and tacky. For many of my choices I would seem to have to use capitalised naming to give a meaningful name. The problem here is these names are used in other Maple packages. So sooner or later there is going to be a clash.

Is there a way a of handling this? Like can I defiine package parameter names? Or should I just stick with my abbreviations?

I read somewhere about this problem in the help years ago. I think Basis was the example used in different packages GroebinerBasis and Polynomial package and using PackageName:-Basis to avoid the clash with both loaded. But that is more a command level handling.

restart

 

illegal:=proc(x,y,{Point:=[symbol=solidcircle,colour=purple]},{Line:=[colour=green,thickness=2]},{Colour:="Blue"},{Scale:=5},{Print:="y"})

print("1 Point= ",Point);  #I currently use points
print("2 Line= ",Line);   #                 linetype
print("3 Colour=  ",Colour); #              clr     this has nothing to do with plotting colour
print("4 Scale= ",Scale);   #               scl   
print("5 Print= ",Print);   #               prnt
Scale*x/y
end proc

 

proc (x, y, { Colour := "Blue", Line := [colour = green, thickness = 2], Point := [symbol = solidcircle, colour = purple], Print := "y", Scale := 5 }) print("1 Point= ", Point); print("2 Line= ", Line); print("3 Colour=  ", Colour); print("4 Scale= ", Scale); print("5 Print= ", Print); Scale*x/y end proc

(1)

illegal(3,7)

"1 Point= ", [symbol = solidcircle, colour = purple]

 

"2 Line= ", [colour = green, thickness = 2]

 

"3 Colour=  ", "Blue"

 

"4 Scale= ", 5

 

"5 Print= ", "y"

 

15/7

(2)

illegal(3,7,line=[linestyle=dash,colour=black,thickness=4])

"1 Point= ", [symbol = solidcircle, colour = purple]

 

"2 Line= ", [colour = green, thickness = 2]

 

"3 Colour=  ", "Blue"

 

"4 Scale= ", 5

 

"5 Print= ", "y"

 

15/7

(3)
 

 

Download 2024-03-09_Illegal_or_Not.mw

I need admin's help
I use evalf(3*21/100,3)=0.630
and evalf(3*89/100,3)=2.67
Is there a way for me to get 2 decimal places
so evalf(3*21/100,3)=0.63?

My apologies for such a basic question. But when I do: my_function:=int((t/τ)*exp(-t/τ),t=0..infinity), I get: lim_{t-->∞} (exp(-t/τ))*τ-(t*exp(-t/τ)) + τ, which with simplify I can get to: lim_{t-->∞} (-t + τ)*(exp(-t/τ))+τ

But why doesn't Maple automatically solve this limit to τ? What do I need to do more?

Thank you for your patient help! Mischa

I very welcome this new feature Thumbs up - Free signs icons but here I am stuck

"restart; f(x):=( sin(x))/(x) :  plot(f(x),title=f(x))"

 

solve(f(x) = 0, x, allsolutions); about(_Z1)

Originally _Z1, renamed _Z1~:
  is assumed to be: integer
 

 

SolveTools:-DisplaySolutions(%)

%PIECEWISE([2*Pi*_Z1, ``])

(1)

Questions:
Q1: How to get also the uneven multiples of Pi?
Q2: Why is zero not excluded?
Q3: How to get the desirable output "{  Pi n        n in `&Zopf;` , n<>0"

 

DisplaySolutions.mw

 

 

How to interpret the output to limit()?

restart;
local gamma;

gamma

(1)

A := -sigma__v^2*(((-2*gamma^2*sigma__d^4*sigma__e^4 - 16)*sigma__v^6 + (-6*gamma^2*sigma__d^4*sigma__e^6 - 4*gamma^2*sigma__d^2*sigma__e^4 - 48*sigma__e^2)*sigma__v^4 - sigma__e^4*(gamma^4*sigma__d^6*sigma__e^6 + 4*gamma^2*sigma__d^4*sigma__e^4 + 8*gamma^2*sigma__d^2*sigma__e^2 + 48)*sigma__v^2 - 4*gamma^2*sigma__d^2*sigma__e^8 - 16*sigma__e^6)*sqrt(gamma^2*sigma__d^2*sigma__e^4 + 4*sigma__e^2 + 4*sigma__v^2) + (2*sigma__d^2*sigma__v^8 + (12*sigma__d^2*sigma__e^2 + 8)*sigma__v^6 + 2*(12 + gamma^2*sigma__d^4*sigma__e^4 + sigma__d^2*(gamma^2 + 13)*sigma__e^2)*sigma__e^2*sigma__v^4 + 8*(3 + gamma^2*sigma__d^4*sigma__e^4 + sigma__d^2*(gamma^2 + 6)*sigma__e^2/2)*sigma__e^4*sigma__v^2 + sigma__e^6*(gamma^2*sigma__d^2*sigma__e^2 + 4)*(gamma^2*sigma__d^4*sigma__e^4 + 2*sigma__d^2*sigma__e^2 + 2))*sigma__v^2*gamma*sigma__d)*sigma__d/(4*(sigma__e^2 + sigma__v^2)^2*(gamma^2*sigma__d^2*sigma__e^4 + 4*sigma__e^2 + 4*sigma__v^2)^2);

-(1/4)*sigma__v^2*(((-2*gamma^2*sigma__d^4*sigma__e^4-16)*sigma__v^6+(-6*gamma^2*sigma__d^4*sigma__e^6-4*gamma^2*sigma__d^2*sigma__e^4-48*sigma__e^2)*sigma__v^4-sigma__e^4*(gamma^4*sigma__d^6*sigma__e^6+4*gamma^2*sigma__d^4*sigma__e^4+8*gamma^2*sigma__d^2*sigma__e^2+48)*sigma__v^2-4*gamma^2*sigma__d^2*sigma__e^8-16*sigma__e^6)*(gamma^2*sigma__d^2*sigma__e^4+4*sigma__e^2+4*sigma__v^2)^(1/2)+(2*sigma__d^2*sigma__v^8+(12*sigma__d^2*sigma__e^2+8)*sigma__v^6+2*(12+gamma^2*sigma__d^4*sigma__e^4+sigma__d^2*(gamma^2+13)*sigma__e^2)*sigma__e^2*sigma__v^4+8*(3+gamma^2*sigma__d^4*sigma__e^4+(1/2)*sigma__d^2*(gamma^2+6)*sigma__e^2)*sigma__e^4*sigma__v^2+sigma__e^6*(gamma^2*sigma__d^2*sigma__e^2+4)*(gamma^2*sigma__d^4*sigma__e^4+2*sigma__d^2*sigma__e^2+2))*sigma__v^2*gamma*sigma__d)*sigma__d/((sigma__e^2+sigma__v^2)^2*(gamma^2*sigma__d^2*sigma__e^4+4*sigma__e^2+4*sigma__v^2)^2)

(2)

# Limits

A__0 := limit(A, gamma = 0);
A__inf_wo_assumptions := limit(A, gamma = infinity);
A__inf_with_assumptions := limit(A, gamma = infinity) assuming 0 < sigma__e, 0 < sigma__v, 0 < sigma__d;

(1/2)*sigma__v^2*sigma__d/(sigma__e^2+sigma__v^2)^(1/2)

 

signum(sigma__d^3*sigma__e^2*sigma__v^4*(-sigma__d*sigma__e^2+(sigma__d^2*sigma__e^4)^(1/2))/(sigma__e^2+sigma__v^2)^2)*infinity

 

0

(3)

Download limits_signum.mw

Was experimenting with methods to handle the representation of 3D lines and plotting them. Where I normally use a point and a direction vector to dascribe the line.

With a bit of experimenting I see the element wise operation `+`~  or  `-`~  using prefix notation saves a lot of time converting vector to lists and vice a versa.  Would be interested to know if there are better techniques.

On the plotting side using plot3d Can the colour of the lines be changed individually? Or should I use a different plotting command?

restart

with(plottools):

l:=([2,-3,1],<3,7/9,6>);   # 3d line point + vector

P:=[7,-8,9]

l := [2, `&ndash;`(3), 1], Vector(3, {(1) = 3, (2) = 7/9, (3) = 6})

 

[7, -8, 9]

(1)

pl:=`+`~(lambda*l[2],l[1]); #3d line as vector eqn

 

Vector(3, {(1) = 3*lambda+2, (2) = (7/9)*lambda-3, (3) = 6*lambda+1})

(2)

vnl:=`-`~(pl,P) ; #vector from Point P to 3D line

 

Vector(3, {(1) = 3*lambda-5, (2) = (7/9)*lambda+5, (3) = 6*lambda-8})

(3)

vnl.l[2] assuming `real` ; #dot product of vectors= 0 when perpendicular

 

(3694/81)*lambda-532/9

(4)

sol:=solve( { (4) }, [lambda] )[];

[lambda = 2394/1847]

(5)

intP:=eval(pl,sol)  #intersection point

Vector(3, {(1) = 10876/1847, (2) = -3679/1847, (3) = 16211/1847})

(6)

l2:=P,eval(vnl,sol) ;  #perpendicular 3D line through P

l2 := [7, `&ndash;`(8), 9], Vector(3, {(1) = -2053/1847, (2) = 11097/1847, (3) = -412/1847})

(7)

pl2:=`+`~(lambda*l2[2],l2[1]); #3D line as vector eqn

Vector(3, {(1) = -(2053/1847)*lambda+7, (2) = (11097/1847)*lambda-8, (3) = -(412/1847)*lambda+9})

(8)

plots:-display(plot3d([pl,pl2],lambda=-.5..1.8,thickness=0,colour=[orange,purple],axes=normal,scaling=constrained),
                point(P,colour=blue ,symbolsize=15,symbol=solidsphere),
                point(l[1],colour=green ,symbolsize=15,symbol=solidsphere),
                point(eval(pl,sol),colour=red ,symbolsize=15,symbol=solidsphere),
                arrow(l,0.2, 0.4, 0.1,colour=green),
                arrow(l2,0.2, 0.4, 0.1,colour=blue));

 

  

 


 

Download Perpendicular_3D_lines.mw

I see a number of MaplePrimes questions answered with good examples using prefix notation.  Is there good reference material somewhere that can show me how to make use of prefix notation when using Maple?  It's not always obvious (to me) how to construct maple commands using prefix operators so I typically just use the standard commands.  Understanding Maples prefix notation would help in understanding some of the answers given to questions in this forum.  Thanks.

Question is attached:

I describe in words the problem I want to solve with Maple. I'll need to work with random variables.

I want to compute Var[A+B+C] where A, B, and C are not independent of each other. In particular, I don't know how to compute Cov[A,B], Cov[A,C], and Cov[B,C]. The model specifications follow.

Let:

A = X__1*(-lambda__1*X__1-lambda__1*delta__1+nu__1-nu__01);
B = X__2*(-lambda__2*X__2-lambda__2*delta__2+nu__2-nu__02);
A = X__3*(-lambda__3*X__3-lambda__3*delta__3+nu__1+nu__2-nu__01-nu__02);

A = X__1*(-X__1*lambda__1-delta__1*lambda__1-nu__01+nu__1)

 

B = X__2*(-X__2*lambda__2-delta__2*lambda__2-nu__02+nu__2)

 

A = X__3*(-X__3*lambda__3-delta__3*lambda__3-nu__01-nu__02+nu__1+nu__2)

(1)

where lambda__1, lambda__2, and lambda__3 are constants. Moreover, nu__01 is the mean of nu__1~N(nu__01,sigma__nu^2)
and nu__02 is the mean of nu__2~N(nu__02,sigma__nu^2). Note that nu__1 and nu__2 have the same variance and are independent of each other.

In addition:

X__1 = beta__1*(nu__1+nu__2-nu__01-nu__02)+alpha__1*delta__1+alpha__2s*delta__2;
X__2 = beta__2*(nu__1+nu__2-nu__01-nu__02)+alpha__2*delta__2+alpha__1s*delta__1;
X__3 = beta__3*(nu__1+nu__2-nu__01-nu__02)+alpha__3*delta__3;

X__1 = beta__1*(nu__1+nu__2-nu__01-nu__02)+alpha__1*delta__1+alpha__2s*delta__2

 

X__2 = beta__2*(nu__1+nu__2-nu__01-nu__02)+alpha__2*delta__2+alpha__1s*delta__1

 

X__3 = beta__3*(nu__1+nu__2-nu__01-nu__02)+alpha__3*delta__3

(2)

where beta__1, beta__2, beta__3, alpha__1, alpha__2, alpha__3, alpha__1s, alpha__2s are constants. Moreover, delta__1~N(0,sigma__d^2), delta__2~N(0,sigma__d^2), and delta__3~N(0,sigma__d3^2) (note the different variance for delta__3). The variables delta__1, delta__2, and delta__3 are independent of each other. Moreover, nu__1 and nu__2 are independent of delta__1, delta__2, and delta__3.

Now, A, B, C are all products of the form W*Q. In general, Var[W*Q] can be found by applying a formula*** which here reduces to Var[W*Q] = sigma__W^2*sigma__Q^2+(Cov[W,Q])^2, where Cov[W,Q] is simply E[W*Q] since E[W]=0 and E[Q]=0 in my three cases. In short, it's relatively straightforward to find Var[A], Var[B], and Var[C]. However, I don't know about the covariance terms. How to tackle the covariance terms, i.e., Cov[A,B], Cov[A,C], and Cov[B,C]?


***See @whuber's comment in Prof. Dilip Sarwate's answer here https://stats.stackexchange.com/questions/15978/variance-of-product-of-dependent-variables


Perhaps it would be useful to automate the computation of E[A+B+C] as well. However, I managed to compute the expectation by hand, with pen and paper. It would be nice to double check with a script.

Download variance_of_sum_of_products.mw

Want to make a rectangular graph (in the sense of Graph Theory) of 4 copies of the path A-B-C-D i.e. four rows of this path. Once the rectangular or matrix like graph is made its ediges are to be weighted as the matrix position of vertex stating from (1,1) at the upper left vertex i.e. "A". Help please.

Hi everyone! :)

I'm new to Maple, so please forgive me if the solution is obvious, but I could not find any fitting articles for my problem.

I have a transfer function and need to find the poles. Later, I want to calculate the constants to meet my filter requirements. For now, I'm focusing on getting a symbolic solution and evaluating the poles.

Here's my setup:


Here is the File NoiseFilter.mw

 

 

When I use the "evalf" function, it gives me a large sum, but I would like to get one complex numerical result.

I appreciate any help! :)

I would expect I could do

A := DataSeries([2, 3, 4], datatype = float)
B := DataSeries([1, 3, 19], datatype = float)
min~(A, B)

but the last command gives me 

Error, (in DataSeries:-min) invalid boolean expression: 2.

How do I find the element-wise min (i.e. DataSeries([1,3,4], datatype=float)) of two DataSeries?

This code used to work OK in Maple 2023.2.1 on windows 10. Now it gives this new internal Maple error 

Error, (in property/ConvertRelation) numeric exception: division by zero

I think this is new problem in Maple 2024 as I have not seen it before. Will report it to Maplesoft. Mean while if someone has any workaround other than not using the `is` command will try it.

The following are two worksheets. One using 2023 shows it works OK (no error) and one in Maple 2024 with the new error

203072

interface(version);

`Standard Worksheet Interface, Maple 2023.2, Windows 10, November 24 2023 Build ID 1762575`

restart;

203072

r:=1/4*a*(a*x^2+6);
tmp:=-1/x^2+1/2*a+(1/x+1/2*a*x)^2;
is(tmp = r)

(1/4)*a*(a*x^2+6)

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

true

 

 

Download no_numeric_exception_march_7_2024_version_2023.mw
 

interface(version);

`Standard Worksheet Interface, Maple 2024.0, Windows 10, March 01 2024 Build ID 1794891`

restart;

r:=1/4*a*(a*x^2+6);
tmp:=-1/x^2+1/2*a+(1/x+1/2*a*x)^2;
is(tmp = r)
 

(1/4)*a*(a*x^2+6)

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

Error, (in property/ConvertRelation) numeric exception: division by zero

 

 

Download numeric_exception_march_7_2024_version_2024.mw

I was wondering if this is by design or not. Here is one example., This ode is solved by Maple 2023.2.1 on windows 10. The solution is in terms of DESol though. It takes less than one minute.

In Maple 2024 on same PC, dsolve hangs on it.

If this by design? Since DESol solution is not very useful actually. I was thinking that may be Maple is now trying to return a solution without DESol in it, and that is why it hangs or take long time. I stopped it after 5 minutes. 

Or should I report it to Maplesoft as possible bug?  

Below are 2 worksheets. One using Maple 2023 and one using 2024.

42444

interface(version);

`Standard Worksheet Interface, Maple 2023.2, Windows 10, November 24 2023 Build ID 1762575`

Physics:-Version();

`The "Physics Updates" version in the MapleCloud is 1684. The version installed in this computer is 1672 created 2024, February 7, 18:34 hours Pacific Time, found in the directory C:\Users\Owner\maple\toolbox\2023\Physics Updates\lib\`

ode:=(x + 1)*diff(u(x), x, x)/((x - 1)*cosh(1/(x - 1))) - (1/((x - 1)*cosh(1/(x - 1))) - (x + 1)/((x - 1)^2*cosh(1/(x - 1))) + (x + 1)*sinh(1/(x - 1))/((x - 1)^3*cosh(1/(x - 1))^2) + (-2*x^3 - 2*x^2)*(x + 1)/((x - 1)^2*cosh(1/(x - 1))^2))*diff(u(x), x) + (x + 1)^2*(x^5 + x^4 + 2*x^2*cosh(1/(x - 1)) - 2*x*cosh(1/(x - 1)) - x - 1)*u(x)/((x - 1)^3*cosh(1/(x - 1))^3);

(x+1)*(diff(diff(u(x), x), x))/((x-1)*cosh(1/(x-1)))-(1/((x-1)*cosh(1/(x-1)))-(x+1)/((x-1)^2*cosh(1/(x-1)))+(x+1)*sinh(1/(x-1))/((x-1)^3*cosh(1/(x-1))^2)+(-2*x^3-2*x^2)*(x+1)/((x-1)^2*cosh(1/(x-1))^2))*(diff(u(x), x))+(x+1)^2*(x^5+x^4+2*x^2*cosh(1/(x-1))-2*x*cosh(1/(x-1))-x-1)*u(x)/((x-1)^3*cosh(1/(x-1))^3)

DEtools:-odeadvisor(ode);

[[_2nd_order, _with_linear_symmetries]]

dsolve(ode); #after about 1 minute...

u(x) = DESol({diff(diff(_Y(x), x), x)-(1/((x-1)*cosh(1/(x-1)))-(x+1)/((x-1)^2*cosh(1/(x-1)))+(x+1)*sinh(1/(x-1))/((x-1)^3*cosh(1/(x-1))^2)+(-2*x^3-2*x^2)*(x+1)/((x-1)^2*cosh(1/(x-1))^2))*(x-1)*cosh(1/(x-1))*(diff(_Y(x), x))/(x+1)+(x+1)*(x^5+x^4+2*x^2*cosh(1/(x-1))-2*x*cosh(1/(x-1))-x-1)*_Y(x)/((x-1)^2*cosh(1/(x-1))^2)}, {_Y(x)})

 

 

Download dsolve_march_7_2024_maple_2023.mw


restart;

interface(version);

`Standard Worksheet Interface, Maple 2024.0, Windows 10, March 01 2024 Build ID 1794891`

ode:=(x + 1)*diff(u(x), x, x)/((x - 1)*cosh(1/(x - 1))) - (1/((x - 1)*cosh(1/(x - 1))) - (x + 1)/((x - 1)^2*cosh(1/(x - 1))) + (x + 1)*sinh(1/(x - 1))/((x - 1)^3*cosh(1/(x - 1))^2) + (-2*x^3 - 2*x^2)*(x + 1)/((x - 1)^2*cosh(1/(x - 1))^2))*diff(u(x), x) + (x + 1)^2*(x^5 + x^4 + 2*x^2*cosh(1/(x - 1)) - 2*x*cosh(1/(x - 1)) - x - 1)*u(x)/((x - 1)^3*cosh(1/(x - 1))^3);

(x+1)*(diff(diff(u(x), x), x))/((x-1)*cosh(1/(x-1)))-(1/((x-1)*cosh(1/(x-1)))-(x+1)/((x-1)^2*cosh(1/(x-1)))+(x+1)*sinh(1/(x-1))/((x-1)^3*cosh(1/(x-1))^2)+(-2*x^3-2*x^2)*(x+1)/((x-1)^2*cosh(1/(x-1))^2))*(diff(u(x), x))+(x+1)^2*(x^5+x^4+2*x^2*cosh(1/(x-1))-2*x*cosh(1/(x-1))-x-1)*u(x)/((x-1)^3*cosh(1/(x-1))^3)

DEtools:-odeadvisor(ode);

[[_2nd_order, _with_linear_symmetries]]

dsolve(ode); #hangs

 

 

Download dsolve_march_7_2024_maple_2024.mw

One may easily spot that the symbols “” and “” are missing here and here

These two PDF files were printed officially, so I think they can be considered examples of best practice for the export functionality. Does this mean that this functionality is still defective in the most recent release?

I have an expression Z_val which I would like to rewrite in a compact, more readable form. Assuming that simplify() did its job, I thought of using (1) alias(), and (2) eval() with some combination of my parameters, since such combination occurs repetitively in my expression.

(1) is not particularly useful for my problem at hand. In particular, powers are not recognized and the alias I defined did not substitute the combination in all the places I was expecting to.

(2) is somehow effective but my expression becomes arguably even more complicated.

What else would you suggest?

EDIT:

restart;

local gamma;

gamma

(1)

M := -lambda*(beta^2*(sigma__v^2+sigma__e^2)+alpha*sigma__d^2*(alpha+1))+beta*sigma__v^2;
V := simplify((1+(Cov__XY/(sigma__X*sigma__Y))^2)*(sigma__X*sigma__Y)^2);

-lambda*(beta^2*(sigma__e^2+sigma__v^2)+alpha*sigma__d^2*(alpha+1))+beta*sigma__v^2

 

sigma__X^2*sigma__Y^2+Cov__XY^2

(2)

# Specify the terms, calculated by hand

sigma__X := beta^2*(sigma__v^2+sigma__e^2)+alpha^2*sigma__d^2;
sigma__Y := lambda^2*(beta^2*(sigma__v^2+sigma__e^2)+sigma__d^2*(alpha+1)^2)+sigma__v^2*(1-2*lambda*beta);
Cov__XY := M;

beta^2*(sigma__e^2+sigma__v^2)+alpha^2*sigma__d^2

 

lambda^2*(beta^2*(sigma__e^2+sigma__v^2)+sigma__d^2*(alpha+1)^2)+sigma__v^2*(-2*beta*lambda+1)

 

-lambda*(beta^2*(sigma__e^2+sigma__v^2)+alpha*sigma__d^2*(alpha+1))+beta*sigma__v^2

(3)

V;

(beta^2*(sigma__e^2+sigma__v^2)+alpha^2*sigma__d^2)^2*(lambda^2*(beta^2*(sigma__e^2+sigma__v^2)+sigma__d^2*(alpha+1)^2)+sigma__v^2*(-2*beta*lambda+1))^2+(-lambda*(beta^2*(sigma__e^2+sigma__v^2)+alpha*sigma__d^2*(alpha+1))+beta*sigma__v^2)^2

(4)

Z := M-(gamma/2)*V;

-lambda*(beta^2*(sigma__e^2+sigma__v^2)+alpha*sigma__d^2*(alpha+1))+beta*sigma__v^2-(1/2)*gamma*((beta^2*(sigma__e^2+sigma__v^2)+alpha^2*sigma__d^2)^2*(lambda^2*(beta^2*(sigma__e^2+sigma__v^2)+sigma__d^2*(alpha+1)^2)+sigma__v^2*(-2*beta*lambda+1))^2+(-lambda*(beta^2*(sigma__e^2+sigma__v^2)+alpha*sigma__d^2*(alpha+1))+beta*sigma__v^2)^2)

(5)

# Specify lambda, beta, and alpha

l := simplify(-gamma*sigma__e^2*sigma__v^2/(2*(sigma__e^2 + sigma__v^2)) + sigma__v^2*sqrt(gamma^2*sigma__d^2*sigma__e^4 + 4*sigma__e^2 + 4*sigma__v^2)/(2*(sigma__e^2 + sigma__v^2)*sigma__d));
b := simplify(eval(sigma__v^2/((gamma*sigma__e^2 + 2*lambda)*sigma__v^2 + 2*lambda*sigma__e^2), lambda = l));
a := simplify(eval(-lambda*(sigma__e^2 + sigma__v^2)/((gamma*sigma__v^2 + 2*lambda)*sigma__e^2 + 2*lambda*sigma__v^2), lambda = l));

(1/2)*sigma__v^2*(-gamma*sigma__e^2*sigma__d+(gamma^2*sigma__d^2*sigma__e^4+4*sigma__e^2+4*sigma__v^2)^(1/2))/((sigma__e^2+sigma__v^2)*sigma__d)

 

sigma__d/(gamma^2*sigma__d^2*sigma__e^4+4*sigma__e^2+4*sigma__v^2)^(1/2)

 

(1/2)*(gamma*sigma__e^2*sigma__d-(gamma^2*sigma__d^2*sigma__e^4+4*sigma__e^2+4*sigma__v^2)^(1/2))/(gamma^2*sigma__d^2*sigma__e^4+4*sigma__e^2+4*sigma__v^2)^(1/2)

(6)

Z_val := simplify(eval(Z, [lambda = l, beta = b, alpha = a]));

-(1/4)*sigma__d*sigma__v^2*(((-2*gamma^2*sigma__d^4*sigma__e^4-16)*sigma__v^6+(-6*gamma^2*sigma__d^4*sigma__e^6-4*gamma^2*sigma__d^2*sigma__e^4-48*sigma__e^2)*sigma__v^4-sigma__e^4*(gamma^4*sigma__d^6*sigma__e^6+4*gamma^2*sigma__d^4*sigma__e^4+8*gamma^2*sigma__d^2*sigma__e^2+48)*sigma__v^2-4*gamma^2*sigma__d^2*sigma__e^8-16*sigma__e^6)*(gamma^2*sigma__d^2*sigma__e^4+4*sigma__e^2+4*sigma__v^2)^(1/2)+(2*sigma__d^2*sigma__v^8+(12*sigma__d^2*sigma__e^2+8)*sigma__v^6+2*(12+gamma^2*sigma__d^4*sigma__e^4+sigma__d^2*(gamma^2+13)*sigma__e^2)*sigma__e^2*sigma__v^4+8*(3+gamma^2*sigma__d^4*sigma__e^4+(1/2)*sigma__d^2*(gamma^2+6)*sigma__e^2)*sigma__e^4*sigma__v^2+sigma__e^6*(gamma^2*sigma__d^2*sigma__e^2+4)*(gamma^2*sigma__d^4*sigma__e^4+2*sigma__d^2*sigma__e^2+2))*sigma__d*sigma__v^2*gamma)/((sigma__e^2+sigma__v^2)^2*(gamma^2*sigma__d^2*sigma__e^4+4*sigma__e^2+4*sigma__v^2)^2)

(7)

eval(Z_val, gamma = Gamma/sigma__e/sigma__d);

-(1/4)*sigma__d*sigma__v^2*(((-2*Gamma^2*sigma__d^2*sigma__e^2-16)*sigma__v^6+(-6*Gamma^2*sigma__d^2*sigma__e^4-4*Gamma^2*sigma__e^2-48*sigma__e^2)*sigma__v^4-sigma__e^4*(Gamma^4*sigma__d^2*sigma__e^2+4*Gamma^2*sigma__d^2*sigma__e^2+8*Gamma^2+48)*sigma__v^2-4*Gamma^2*sigma__e^6-16*sigma__e^6)*(Gamma^2*sigma__e^2+4*sigma__e^2+4*sigma__v^2)^(1/2)+(2*sigma__d^2*sigma__v^8+(12*sigma__d^2*sigma__e^2+8)*sigma__v^6+2*(12+Gamma^2*sigma__e^2*sigma__d^2+sigma__d^2*(Gamma^2/(sigma__e^2*sigma__d^2)+13)*sigma__e^2)*sigma__e^2*sigma__v^4+8*(3+Gamma^2*sigma__e^2*sigma__d^2+(1/2)*sigma__d^2*(Gamma^2/(sigma__e^2*sigma__d^2)+6)*sigma__e^2)*sigma__e^4*sigma__v^2+sigma__e^6*(Gamma^2+4)*(Gamma^2*sigma__d^2*sigma__e^2+2*sigma__d^2*sigma__e^2+2))*sigma__v^2*Gamma/sigma__e)/((sigma__e^2+sigma__v^2)^2*(Gamma^2*sigma__e^2+4*sigma__e^2+4*sigma__v^2)^2)

(8)

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