Mark1993

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These are questions asked by Mark1993

Good evening!!!

Let me briefly describe the problem I've faced recently.

The program (attached) deals with a rather complicated function f depending on parametrs eps1, eps2, eps3, eps4 and variable w. The aim is to expand the function f(w1) into Taylor series with respect to all parametrs (eps1, eps2, eps3, eps4) in order to study its asymptotic behavior as function depending only on k; 0<k<1.

I decided to use mtaylor-function for that problem, which (as I've understood) is the only one to be applied in such cases, but the result was rather unsatisfactory, an error: 

Error, (in gcd/LinZip) input must be polynomials over the integers

Programm code: (1)-(12) only announcing functions....(((, see below
 

f := proc (w) options operator, arrow; -B1+(A1-C1)*w+(B1-D1)*w^2-A1*w^3 end proc

proc (w) options operator, arrow; -B1+(A1-C1)*w+(B1-D1)*w^2-A1*w^3 end proc

(1)

f1 := proc (w) options operator, arrow; A1-C1+(2*B1-2*D1)*w-3*A1*w^2 end proc

proc (w) options operator, arrow; A1-C1+(2*B1-2*D1)*w-3*A1*w^2 end proc

(2)

w1 := (B1-D1+sqrt((B1-D1)^2+3*A1*(A1-C1)))/(3*A1)

(1/3)*(B1-D1+(B1^2-2*B1*D1+D1^2+3*A1^2-3*A1*C1)^(1/2))/A1

(3)

f(w1)

-B1+(1/3)*(A1-C1)*(B1-D1+(B1^2-2*B1*D1+D1^2+3*A1^2-3*A1*C1)^(1/2))/A1+(1/9)*(B1-D1)*(B1-D1+(B1^2-2*B1*D1+D1^2+3*A1^2-3*A1*C1)^(1/2))^2/A1^2-(1/27)*(B1-D1+(B1^2-2*B1*D1+D1^2+3*A1^2-3*A1*C1)^(1/2))^3/A1^2

(4)

s := eps4*sin(l*tau)+(4*(l*sqrt(k/(1-k))+l*eps3)+2*l*((1-2*k)/sqrt(k*(1-k))+eps1))/l^2

eps4*sin(l*tau)+(4*l*(k/(1-k))^(1/2)+4*l*eps3+2*l*((1-2*k)/(k*(1-k))^(1/2)+eps1))/l^2

(5)

A1 := (2*(l*sqrt(k/(1-k))+l*eps3)+l*((1-2*k)/sqrt(k*(1-k))+eps1))/s

(2*l*(k/(1-k))^(1/2)+2*l*eps3+l*((1-2*k)/(k*(1-k))^(1/2)+eps1))/(eps4*sin(l*tau)+(4*l*(k/(1-k))^(1/2)+4*l*eps3+2*l*((1-2*k)/(k*(1-k))^(1/2)+eps1))/l^2)

(6)

A1 := (2*(l*sqrt(k/(1-k))+l*eps3)+l*((1-2*k)/sqrt(k*(1-k))+eps1))/s

(2*l*(k/(1-k))^(1/2)+2*l*eps3+l*((1-2*k)/(k*(1-k))^(1/2)+eps1))/(eps4*sin(l*tau)+(4*l*(k/(1-k))^(1/2)+4*l*eps3+2*l*((1-2*k)/(k*(1-k))^(1/2)+eps1))/l^2)

(7)

B1 := 4/s^2

4/(eps4*sin(l*tau)+(4*l*(k/(1-k))^(1/2)+4*l*eps3+2*l*((1-2*k)/(k*(1-k))^(1/2)+eps1))/l^2)^2

(8)

C1 := (((1-2*k)/sqrt(k*(1-k))+eps1)^2+(-(1-2*k)/sqrt(k*(1-k))+eps2)^2)/s^2

(((1-2*k)/(k*(1-k))^(1/2)+eps1)^2+(-(1-2*k)/(k*(1-k))^(1/2)+eps2)^2)/(eps4*sin(l*tau)+(4*l*(k/(1-k))^(1/2)+4*l*eps3+2*l*((1-2*k)/(k*(1-k))^(1/2)+eps1))/l^2)^2

(9)

D1 := (2*((1-2*k)/sqrt(k*(1-k))+eps1))*(-(1-2*k)/sqrt(k*(1-k))+eps2)/s^2

2*((1-2*k)/(k*(1-k))^(1/2)+eps1)*(-(1-2*k)/(k*(1-k))^(1/2)+eps2)/(eps4*sin(l*tau)+(4*l*(k/(1-k))^(1/2)+4*l*eps3+2*l*((1-2*k)/(k*(1-k))^(1/2)+eps1))/l^2)^2

(10)

l := 1

1

(11)

f(w1)

-4/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2+(1/3)*((2*(k/(1-k))^(1/2)+2*eps3+(1-2*k)/(k*(1-k))^(1/2)+eps1)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)-(((1-2*k)/(k*(1-k))^(1/2)+eps1)^2+(-(1-2*k)/(k*(1-k))^(1/2)+eps2)^2)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2)*(4/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2-2*((1-2*k)/(k*(1-k))^(1/2)+eps1)*(-(1-2*k)/(k*(1-k))^(1/2)+eps2)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2+(16/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^4-16*((1-2*k)/(k*(1-k))^(1/2)+eps1)*(-(1-2*k)/(k*(1-k))^(1/2)+eps2)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^4+4*((1-2*k)/(k*(1-k))^(1/2)+eps1)^2*(-(1-2*k)/(k*(1-k))^(1/2)+eps2)^2/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^4+3*(2*(k/(1-k))^(1/2)+2*eps3+(1-2*k)/(k*(1-k))^(1/2)+eps1)^2/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2-3*(2*(k/(1-k))^(1/2)+2*eps3+(1-2*k)/(k*(1-k))^(1/2)+eps1)*(((1-2*k)/(k*(1-k))^(1/2)+eps1)^2+(-(1-2*k)/(k*(1-k))^(1/2)+eps2)^2)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^3)^(1/2))*(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)/(2*(k/(1-k))^(1/2)+2*eps3+(1-2*k)/(k*(1-k))^(1/2)+eps1)+(1/9)*(4/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2-2*((1-2*k)/(k*(1-k))^(1/2)+eps1)*(-(1-2*k)/(k*(1-k))^(1/2)+eps2)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2)*(4/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2-2*((1-2*k)/(k*(1-k))^(1/2)+eps1)*(-(1-2*k)/(k*(1-k))^(1/2)+eps2)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2+(16/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^4-16*((1-2*k)/(k*(1-k))^(1/2)+eps1)*(-(1-2*k)/(k*(1-k))^(1/2)+eps2)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^4+4*((1-2*k)/(k*(1-k))^(1/2)+eps1)^2*(-(1-2*k)/(k*(1-k))^(1/2)+eps2)^2/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^4+3*(2*(k/(1-k))^(1/2)+2*eps3+(1-2*k)/(k*(1-k))^(1/2)+eps1)^2/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2-3*(2*(k/(1-k))^(1/2)+2*eps3+(1-2*k)/(k*(1-k))^(1/2)+eps1)*(((1-2*k)/(k*(1-k))^(1/2)+eps1)^2+(-(1-2*k)/(k*(1-k))^(1/2)+eps2)^2)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^3)^(1/2))^2*(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2/(2*(k/(1-k))^(1/2)+2*eps3+(1-2*k)/(k*(1-k))^(1/2)+eps1)^2-(1/27)*(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2*(4/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2-2*((1-2*k)/(k*(1-k))^(1/2)+eps1)*(-(1-2*k)/(k*(1-k))^(1/2)+eps2)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2+(16/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^4-16*((1-2*k)/(k*(1-k))^(1/2)+eps1)*(-(1-2*k)/(k*(1-k))^(1/2)+eps2)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^4+4*((1-2*k)/(k*(1-k))^(1/2)+eps1)^2*(-(1-2*k)/(k*(1-k))^(1/2)+eps2)^2/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^4+3*(2*(k/(1-k))^(1/2)+2*eps3+(1-2*k)/(k*(1-k))^(1/2)+eps1)^2/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^2-3*(2*(k/(1-k))^(1/2)+2*eps3+(1-2*k)/(k*(1-k))^(1/2)+eps1)*(((1-2*k)/(k*(1-k))^(1/2)+eps1)^2+(-(1-2*k)/(k*(1-k))^(1/2)+eps2)^2)/(eps4*sin(tau)+4*(k/(1-k))^(1/2)+4*eps3+2*(1-2*k)/(k*(1-k))^(1/2)+2*eps1)^3)^(1/2))^3/(2*(k/(1-k))^(1/2)+2*eps3+(1-2*k)/(k*(1-k))^(1/2)+eps1)^2

(12)

assume(0 < k and k < 1)

mtaylor(f(w1), [eps1, eps2, eps3, eps4], 2)

Error, (in gcd/LinZip) input must be polynomials over the integers

 

``


Wish you could give some advice on how to improve the situation.

Thanks a lot in advance.

Download res2.mw

 

 

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