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Question: How by apply limit remove the constant with different index?

Question:How by apply limit remove the constant with different index?

salim-barzani 655 Maple 2024
limit duplicate-question
February 16 2025
2

How apply long wave limit for removing the constant k in such function , i need a general formula 

Limiting process from eq 12 to Bij

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restart
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NULL

Eq 12.

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eij := ((-3*k[i]*(k[i]-k[j])*l[j]+beta)*l[i]^2-(2*(-3*k[j]*(k[i]-k[j])*l[j]*(1/2)+beta))*l[j]*l[i]+beta*l[j]^2)/((-3*k[i]*(k[i]+k[j])*l[j]+beta)*l[i]^2-(2*(3*k[j]*(k[i]+k[j])*l[j]*(1/2)+beta))*l[j]*l[i]+beta*l[j]^2)

((-3*k[i]*(k[i]-k[j])*l[j]+beta)*l[i]^2-2*(-(3/2)*k[j]*(k[i]-k[j])*l[j]+beta)*l[j]*l[i]+beta*l[j]^2)/((-3*k[i]*(k[i]+k[j])*l[j]+beta)*l[i]^2-2*((3/2)*k[j]*(k[i]+k[j])*l[j]+beta)*l[j]*l[i]+beta*l[j]^2)

 (1)
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eval(eij, k[j] = k[i]); series(%, k[i], 3); convert(%, polynom); eval(%, k[j] = k[i]); Bij := %

(beta*l[i]^2-2*beta*l[i]*l[j]+beta*l[j]^2)/((-6*k[i]^2*l[j]+beta)*l[i]^2-2*(3*k[i]^2*l[j]+beta)*l[j]*l[i]+beta*l[j]^2)

  

series(1+((6*l[i]^2*l[j]+6*l[i]*l[j]^2)/(beta*l[i]^2-2*beta*l[i]*l[j]+beta*l[j]^2))*k[i]^2+O(k[i]^4),k[i],4)

  

1+(6*l[i]^2*l[j]+6*l[i]*l[j]^2)*k[i]^2/(beta*l[i]^2-2*beta*l[i]*l[j]+beta*l[j]^2)

  

1+(6*l[i]^2*l[j]+6*l[i]*l[j]^2)*k[i]^2/(beta*l[i]^2-2*beta*l[i]*l[j]+beta*l[j]^2)

  

1+(6*l[i]^2*l[j]+6*l[i]*l[j]^2)*k[i]^2/(beta*l[i]^2-2*beta*l[i]*l[j]+beta*l[j]^2)

 (2)
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NULL

NULL

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