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How to simplify the arguments of the exp...

FZ 30 Maple 2022
simplify simplification trigh
March 07 2025
2 2

How do we simplify the arguments of the exponential in (1)? Further how to express (1) into hyperbolic/trig functions? 
 

> 

restart
> 

with(LinearAlgebra)
> 

Bans := -8*delta2*delta4*delta1*(delta3^2+delta4^2)*exp((-(2*I)*y*a*(delta3^2+delta4^2)*delta1^3+((2*I)*y*delta3^3*a+(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+((2*I)*y*delta4^2*a-I*t)*delta3+delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta1^2-(2*I)*(y*delta2^2*a-(1/2)*t)*(delta3^2+delta4^2)*delta1+(2*(I*y*delta3^3*a+(1/2)*(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-I*t*(1/2)+I*y*delta4^2*a)*delta3+(1/2)*delta4^2*(x+y+t)^2*(B2+B1+B3+B4)))*delta2^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))-8*delta2*delta3*delta4*(delta1^2+delta2^2)*exp(((2*I)*y*a*(delta3^2+delta4^2)*delta1^3+(-(2*I)*y*delta3^3*a+(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-(2*I)*y*delta4^2*a+I*t)*delta3+delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta1^2+(2*I)*(y*delta2^2*a-(1/2)*t)*(delta3^2+delta4^2)*delta1-(2*(I*y*delta3^3*a-(1/2)*(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-I*t*(1/2)+I*y*delta4^2*a)*delta3-(1/2)*delta4^2*(x+y+t)^2*(B2+B1+B3+B4)))*delta2^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))+(delta2^2-2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((delta3+I*delta4)*delta2^2+(I*delta3^2+I*delta4^2)*delta2+delta1*(delta4^2+I*delta4*delta1+delta3*(delta3+delta1)))*exp((-2*y*a*(delta3^2+delta4^2)*delta2^3+(-2*y*delta4^3*a+2*(x+y+t)^2*(B1+B3)*delta4^2+(-2*a*delta3^2*y-t)*delta4+2*delta3^2*(x+y+t)^2*(B1+B3))*delta2^2-(2*(delta3^2+delta4^2))*(y*a*delta1^2+(1/2)*t)*delta2+2*delta1^2*(-y*delta4^3*a+(x+y+t)^2*(B1+B3)*delta4^2+(-(1/2)*t-y*a*delta3^2)*delta4+delta3^2*(x+y+t)^2*(B1+B3)))/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))+(delta2^2+2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((delta3+I*delta4)*delta2^2+(-I*delta3^2-I*delta4^2)*delta2+delta1*(delta4^2+I*delta4*delta1+delta3*(delta3+delta1)))*exp((2*y*a*(delta3^2+delta4^2)*delta2^3+(-2*y*delta4^3*a+2*(x+y+t)^2*(B2+B3)*delta4^2+(-2*a*delta3^2*y-t)*delta4+2*delta3^2*(x+y+t)^2*(B2+B3))*delta2^2+(2*(delta3^2+delta4^2))*(y*a*delta1^2+(1/2)*t)*delta2+2*delta1^2*(-y*delta4^3*a+(x+y+t)^2*(B2+B3)*delta4^2+(-(1/2)*t-y*a*delta3^2)*delta4+delta3^2*(x+y+t)^2*(B2+B3)))/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))-(delta2^2+2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((I*delta4-delta3)*delta2^2+(-I*delta3^2-I*delta4^2)*delta2+delta1*(-delta4^2+I*delta4*delta1-delta3*(delta3+delta1)))*exp((-2*y*a*(delta3^2+delta4^2)*delta2^3+(2*y*delta4^3*a+2*(x+y+t)^2*(B1+B4)*delta4^2+(2*a*delta3^2*y+t)*delta4+2*delta3^2*(x+y+t)^2*(B1+B4))*delta2^2-(2*(delta3^2+delta4^2))*(y*a*delta1^2+(1/2)*t)*delta2+(2*(y*delta4^3*a+(x+y+t)^2*(B1+B4)*delta4^2+(y*a*delta3^2+(1/2)*t)*delta4+delta3^2*(x+y+t)^2*(B1+B4)))*delta1^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))-(delta2^2-2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((I*delta4-delta3)*delta2^2+(I*delta3^2+I*delta4^2)*delta2+delta1*(-delta4^2+I*delta4*delta1-delta3*(delta3+delta1)))*exp((2*y*a*(delta3^2+delta4^2)*delta2^3+(2*y*delta4^3*a+2*(x+y+t)^2*(B2+B4)*delta4^2+(2*a*delta3^2*y+t)*delta4+2*delta3^2*(x+y+t)^2*(B2+B4))*delta2^2+(2*(delta3^2+delta4^2))*(y*a*delta1^2+(1/2)*t)*delta2+(2*(y*delta4^3*a+(x+y+t)^2*(B2+B4)*delta4^2+(y*a*delta3^2+(1/2)*t)*delta4+delta3^2*(x+y+t)^2*(B2+B4)))*delta1^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))

-8*delta2*delta4*delta1*(delta3^2+delta4^2)*exp((-(2*I)*y*a*(delta3^2+delta4^2)*delta1^3+((2*I)*y*delta3^3*a+(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+((2*I)*y*delta4^2*a-I*t)*delta3+delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta1^2-(2*I)*(y*delta2^2*a-(1/2)*t)*(delta3^2+delta4^2)*delta1+2*(I*y*delta3^3*a+(1/2)*(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-((1/2)*I)*t+I*y*delta4^2*a)*delta3+(1/2)*delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta2^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))-8*delta2*delta3*delta4*(delta1^2+delta2^2)*exp(((2*I)*y*a*(delta3^2+delta4^2)*delta1^3+(-(2*I)*y*delta3^3*a+(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-(2*I)*y*delta4^2*a+I*t)*delta3+delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta1^2+(2*I)*(y*delta2^2*a-(1/2)*t)*(delta3^2+delta4^2)*delta1-2*(I*y*delta3^3*a-(1/2)*(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-((1/2)*I)*t+I*y*delta4^2*a)*delta3-(1/2)*delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta2^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))+(delta2^2-2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((delta3+I*delta4)*delta2^2+(I*delta3^2+I*delta4^2)*delta2+delta1*(delta4^2+I*delta4*delta1+delta3*(delta3+delta1)))*exp((-2*y*a*(delta3^2+delta4^2)*delta2^3+(-2*y*delta4^3*a+2*(x+y+t)^2*(B1+B3)*delta4^2+(-2*a*delta3^2*y-t)*delta4+2*delta3^2*(x+y+t)^2*(B1+B3))*delta2^2-2*(delta3^2+delta4^2)*(y*a*delta1^2+(1/2)*t)*delta2+2*delta1^2*(-y*delta4^3*a+(x+y+t)^2*(B1+B3)*delta4^2+(-(1/2)*t-y*a*delta3^2)*delta4+delta3^2*(x+y+t)^2*(B1+B3)))/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))+(delta2^2+2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((delta3+I*delta4)*delta2^2+(-I*delta3^2-I*delta4^2)*delta2+delta1*(delta4^2+I*delta4*delta1+delta3*(delta3+delta1)))*exp((2*y*a*(delta3^2+delta4^2)*delta2^3+(-2*y*delta4^3*a+2*(x+y+t)^2*(B2+B3)*delta4^2+(-2*a*delta3^2*y-t)*delta4+2*delta3^2*(x+y+t)^2*(B2+B3))*delta2^2+2*(delta3^2+delta4^2)*(y*a*delta1^2+(1/2)*t)*delta2+2*delta1^2*(-y*delta4^3*a+(x+y+t)^2*(B2+B3)*delta4^2+(-(1/2)*t-y*a*delta3^2)*delta4+delta3^2*(x+y+t)^2*(B2+B3)))/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))-(delta2^2+2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((I*delta4-delta3)*delta2^2+(-I*delta3^2-I*delta4^2)*delta2+delta1*(-delta4^2+I*delta4*delta1-delta3*(delta3+delta1)))*exp((-2*y*a*(delta3^2+delta4^2)*delta2^3+(2*y*delta4^3*a+2*(x+y+t)^2*(B1+B4)*delta4^2+(2*a*delta3^2*y+t)*delta4+2*delta3^2*(x+y+t)^2*(B1+B4))*delta2^2-2*(delta3^2+delta4^2)*(y*a*delta1^2+(1/2)*t)*delta2+2*(y*delta4^3*a+(x+y+t)^2*(B1+B4)*delta4^2+(y*a*delta3^2+(1/2)*t)*delta4+delta3^2*(x+y+t)^2*(B1+B4))*delta1^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))-(delta2^2-2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((I*delta4-delta3)*delta2^2+(I*delta3^2+I*delta4^2)*delta2+delta1*(-delta4^2+I*delta4*delta1-delta3*(delta3+delta1)))*exp((2*y*a*(delta3^2+delta4^2)*delta2^3+(2*y*delta4^3*a+2*(x+y+t)^2*(B2+B4)*delta4^2+(2*a*delta3^2*y+t)*delta4+2*delta3^2*(x+y+t)^2*(B2+B4))*delta2^2+2*(delta3^2+delta4^2)*(y*a*delta1^2+(1/2)*t)*delta2+2*(y*delta4^3*a+(x+y+t)^2*(B2+B4)*delta4^2+(y*a*delta3^2+(1/2)*t)*delta4+delta3^2*(x+y+t)^2*(B2+B4))*delta1^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))

 (1)
> 

argexp1 := simplify((-(2*I)*y*a*(delta3^2+delta4^2)*delta1^3+((2*I)*y*delta3^3*a+(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+((2*I)*y*delta4^2*a-I*t)*delta3+delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta1^2-(2*I)*(y*delta2^2*a-(1/2)*t)*(delta3^2+delta4^2)*delta1+(2*(I*y*delta3^3*a+(1/2)*(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-I*t*(1/2)+I*y*delta4^2*a)*delta3+(1/2)*delta4^2*(x+y+t)^2*(B2+B1+B3+B4)))*delta2^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)), size)

(-(2*I)*y*a*(delta3^2+delta4^2)*delta1^3+((2*I)*y*delta3^3*a+(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+((2*I)*y*delta4^2*a-I*t)*delta3+delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta1^2-(2*I)*(y*delta2^2*a-(1/2)*t)*(delta3^2+delta4^2)*delta1+2*(y*delta3^3*a*I+(1/2)*(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-((1/2)*I)*t+y*delta4^2*a*I)*delta3+(1/2)*delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta2^2)/((delta1+delta2*I)*(delta4*I-delta3)*(delta3+delta4*I)*(delta2*I-delta1))

 (2)
> 

 

> 

argexp2 := ((2*I)*y*a*(delta3^2+delta4^2)*delta1^3+(-(2*I)*y*delta3^3*a+(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-(2*I)*y*delta4^2*a+I*t)*delta3+delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta1^2+(2*I)*(y*delta2^2*a-(1/2)*t)*(delta3^2+delta4^2)*delta1-(2*(I*y*delta3^3*a-(1/2)*(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-I*t*(1/2)+I*y*delta4^2*a)*delta3-(1/2)*delta4^2*(x+y+t)^2*(B2+B1+B3+B4)))*delta2^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1))

((2*I)*y*a*(delta3^2+delta4^2)*delta1^3+(-(2*I)*y*delta3^3*a+(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-(2*I)*y*delta4^2*a+t*I)*delta3+delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta1^2+(2*I)*(y*delta2^2*a-(1/2)*t)*(delta3^2+delta4^2)*delta1-2*(y*delta3^3*a*I-(1/2)*(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-((1/2)*I)*t+y*delta4^2*a*I)*delta3-(1/2)*delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta2^2)/((delta1+delta2*I)*(delta4*I-delta3)*(delta3+delta4*I)*(delta2*I-delta1))

 (3)
> 

simplify(argexp1+argexp2)

2*(x+y+t)^2*(B2+B1+B3+B4)

 (4)
> 

terms := op(Bans)

-8*delta2*delta4*delta1*(delta3^2+delta4^2)*exp((-(2*I)*y*a*(delta3^2+delta4^2)*delta1^3+((2*I)*y*delta3^3*a+(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+((2*I)*y*delta4^2*a-I*t)*delta3+delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta1^2-(2*I)*(y*delta2^2*a-(1/2)*t)*(delta3^2+delta4^2)*delta1+2*(I*y*delta3^3*a+(1/2)*(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-((1/2)*I)*t+I*y*delta4^2*a)*delta3+(1/2)*delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta2^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1))), -8*delta2*delta3*delta4*(delta1^2+delta2^2)*exp(((2*I)*y*a*(delta3^2+delta4^2)*delta1^3+(-(2*I)*y*delta3^3*a+(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-(2*I)*y*delta4^2*a+I*t)*delta3+delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta1^2+(2*I)*(y*delta2^2*a-(1/2)*t)*(delta3^2+delta4^2)*delta1-2*(I*y*delta3^3*a-(1/2)*(x+y+t)^2*(B2+B1+B3+B4)*delta3^2+(-((1/2)*I)*t+I*y*delta4^2*a)*delta3-(1/2)*delta4^2*(x+y+t)^2*(B2+B1+B3+B4))*delta2^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1))), (delta2^2-2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((delta3+I*delta4)*delta2^2+(I*delta3^2+I*delta4^2)*delta2+delta1*(delta4^2+I*delta4*delta1+delta3*(delta3+delta1)))*exp((-2*y*a*(delta3^2+delta4^2)*delta2^3+(-2*y*delta4^3*a+2*(x+y+t)^2*(B1+B3)*delta4^2+(-2*a*delta3^2*y-t)*delta4+2*delta3^2*(x+y+t)^2*(B1+B3))*delta2^2-2*(delta3^2+delta4^2)*(y*a*delta1^2+(1/2)*t)*delta2+2*delta1^2*(-y*delta4^3*a+(x+y+t)^2*(B1+B3)*delta4^2+(-(1/2)*t-y*a*delta3^2)*delta4+delta3^2*(x+y+t)^2*(B1+B3)))/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1))), (delta2^2+2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((delta3+I*delta4)*delta2^2+(-I*delta3^2-I*delta4^2)*delta2+delta1*(delta4^2+I*delta4*delta1+delta3*(delta3+delta1)))*exp((2*y*a*(delta3^2+delta4^2)*delta2^3+(-2*y*delta4^3*a+2*(x+y+t)^2*(B2+B3)*delta4^2+(-2*a*delta3^2*y-t)*delta4+2*delta3^2*(x+y+t)^2*(B2+B3))*delta2^2+2*(delta3^2+delta4^2)*(y*a*delta1^2+(1/2)*t)*delta2+2*delta1^2*(-y*delta4^3*a+(x+y+t)^2*(B2+B3)*delta4^2+(-(1/2)*t-y*a*delta3^2)*delta4+delta3^2*(x+y+t)^2*(B2+B3)))/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1))), -(delta2^2+2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((I*delta4-delta3)*delta2^2+(-I*delta3^2-I*delta4^2)*delta2+delta1*(-delta4^2+I*delta4*delta1-delta3*(delta3+delta1)))*exp((-2*y*a*(delta3^2+delta4^2)*delta2^3+(2*y*delta4^3*a+2*(x+y+t)^2*(B1+B4)*delta4^2+(2*a*delta3^2*y+t)*delta4+2*delta3^2*(x+y+t)^2*(B1+B4))*delta2^2-2*(delta3^2+delta4^2)*(y*a*delta1^2+(1/2)*t)*delta2+2*(y*delta4^3*a+(x+y+t)^2*(B1+B4)*delta4^2+(y*a*delta3^2+(1/2)*t)*delta4+delta3^2*(x+y+t)^2*(B1+B4))*delta1^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1))), -(delta2^2-2*delta4*delta2+delta4^2+(delta3-delta1)^2)*((I*delta4-delta3)*delta2^2+(I*delta3^2+I*delta4^2)*delta2+delta1*(-delta4^2+I*delta4*delta1-delta3*(delta3+delta1)))*exp((2*y*a*(delta3^2+delta4^2)*delta2^3+(2*y*delta4^3*a+2*(x+y+t)^2*(B2+B4)*delta4^2+(2*a*delta3^2*y+t)*delta4+2*delta3^2*(x+y+t)^2*(B2+B4))*delta2^2+2*(delta3^2+delta4^2)*(y*a*delta1^2+(1/2)*t)*delta2+2*(y*delta4^3*a+(x+y+t)^2*(B2+B4)*delta4^2+(y*a*delta3^2+(1/2)*t)*delta4+delta3^2*(x+y+t)^2*(B2+B4))*delta1^2)/((delta1+I*delta2)*(I*delta4-delta3)*(delta3+I*delta4)*(I*delta2-delta1)))

 (5)
> 

NULL


 

Download argument.mw

Why Doesn’t simplify Fully Simplify My E...

FZ 30 Maple 2022
complex simplify simplification
February 10 2025
3 2

I am working with an expression in Maple that involves complex terms and an integral. After applying the simplify command, some terms remain unsimplified, even though they seem reducible (see (7)). Additionally, an integral in my expression remains unevaluated (see (9)).
 

> 

restart;
> 

kernelopts(version);

`Maple 2022.0, X86 64 WINDOWS, Mar 8 2022, Build ID 1599809`

 (1)
> 

with(plots)
> 

interface(showassumed=0):
> 

assume(x::real);assume(t::real);assume(lambda1::complex);assume(b::real);
> 

alias(psi1 = psi1(x,t), psi2 = psi2(x,t), phi1 = phi1(x,t), phi2 = phi2(x,t), beta = beta(t), alpha =alpha(t));

psi1, psi2, phi1, phi2, beta, alpha

 (2)
> 

rel := {psi1 = exp((-I*lambda1)*x - (1/(4*I*lambda1))*int((alpha + b*beta),t)), psi2 = exp((I*lambda1)*x + (1/(4*I*lambda1))*int((alpha + b*beta),t)), phi1= exp((-I*conjugate(lambda1))*x - (1/(4*I*conjugate(lambda1)))*int((alpha + b*beta),t)), phi2 = exp((I*conjugate(lambda1))*x + (1/(4*I*conjugate(lambda1)))*int((alpha + b*beta),t))}

{phi1 = exp(-I*conjugate(lambda1)*x+((1/4)*I)*(int(b*beta+alpha, t))/conjugate(lambda1)), phi2 = exp(I*conjugate(lambda1)*x-((1/4)*I)*(int(b*beta+alpha, t))/conjugate(lambda1)), psi1 = exp(-I*lambda1*x+((1/4)*I)*(int(b*beta+alpha, t))/lambda1), psi2 = exp(I*lambda1*x-((1/4)*I)*(int(b*beta+alpha, t))/lambda1)}

 (3)
> 

Bnum := psi2*phi1*conjugate(lambda1) + psi1*lambda1*phi2;

psi2*phi1*conjugate(lambda1)+psi1*lambda1*phi2

 (4)
> 

Bnumexp := subs(rel,Bnum):
> 

Den := -phi1*psi2 - phi2*psi1;

-phi1*psi2-phi2*psi1

 (5)
> 

expDen := subs(rel, Den)

-exp(-I*conjugate(lambda1)*x+((1/4)*I)*(int(b*beta+alpha, t))/conjugate(lambda1))*exp(I*lambda1*x-((1/4)*I)*(int(b*beta+alpha, t))/lambda1)-exp(I*conjugate(lambda1)*x-((1/4)*I)*(int(b*beta+alpha, t))/conjugate(lambda1))*exp(-I*lambda1*x+((1/4)*I)*(int(b*beta+alpha, t))/lambda1)

 (6)
> 

sr := Bnumexp/expDen: ratiosr := simplify(diff(sr,t), complex):
> 

B := b - (4*I/beta(t))*ratiosr

b+2*(b*beta+alpha)*exp(((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*beta+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*beta+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*lambda1^2*x-(int(b*beta+alpha, t)))/lambda1)*(-conjugate(lambda1)+lambda1)^2*exp(((1/4)*I)*(4*lambda1^2*x-(int(b*beta+alpha, t)))/lambda1)/(beta(t)*(exp(((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*beta+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*lambda1^2*x-(int(b*beta+alpha, t)))/lambda1)+exp(-((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*beta+alpha, t)))/conjugate(lambda1))*exp(((1/4)*I)*(4*lambda1^2*x-(int(b*beta+alpha, t)))/lambda1))^2*lambda1*conjugate(lambda1))

 (7)
> 

p := {alpha(t) = t^2, beta = exp(-t)}

{beta = exp(-t), alpha(t) = t^2}

 (8)
> 

B1 := eval(subs(p, B))

b+2*(b*exp(-t)+alpha)*exp(((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*exp(-t)+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*exp(-t)+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*lambda1^2*x-(int(b*exp(-t)+alpha, t)))/lambda1)*(-conjugate(lambda1)+lambda1)^2*exp(((1/4)*I)*(4*lambda1^2*x-(int(b*exp(-t)+alpha, t)))/lambda1)/((exp(-t))(t)*(exp(((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*exp(-t)+alpha, t)))/conjugate(lambda1))*exp(-((1/4)*I)*(4*lambda1^2*x-(int(b*exp(-t)+alpha, t)))/lambda1)+exp(-((1/4)*I)*(4*conjugate(lambda1)^2*x-(int(b*exp(-t)+alpha, t)))/conjugate(lambda1))*exp(((1/4)*I)*(4*lambda1^2*x-(int(b*exp(-t)+alpha, t)))/lambda1))^2*lambda1*conjugate(lambda1))

 (9)

NULL


 

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How can I obtain the Hamiltonian?...

FZ 30 Maple 2022
physics differential-equations hamiltonian
February 05 2025
1 8

How can I obtain the Hamiltonian of Eq. (1) in terms of dynamical variables in Maple?

> 

restart
> 

with(PDEtools)
> 

declare(phi(xi), psi(xi))

phi(xi)*`will now be displayed as`*phi

  

psi(xi)*`will now be displayed as`*psi

 (1)
> 

r1 := (1/2*(phi(xi)^2-2*c))*(diff(diff(phi(xi), xi), xi))+((-(1/2)*phi(xi)^2+c)*(k*phi(xi)^2-Omega)^2*(phi(xi)^2-c)^2/(phi(xi)^4*(phi(xi)^2-2*c)^2)-(-k*phi(xi)^2+c*k+Omega)*(k*phi(xi)^2-Omega)*(phi(xi)^2-c)/(phi(xi)^2*(phi(xi)^2-2*c))-(1/2)*k^2*phi(xi)^2+Omega*k+(diff(phi(xi), xi))^2+1)*phi(xi) = 0

(1/2)*(phi(xi)^2-2*c)*(diff(diff(phi(xi), xi), xi))+((-(1/2)*phi(xi)^2+c)*(k*phi(xi)^2-Omega)^2*(phi(xi)^2-c)^2/(phi(xi)^4*(phi(xi)^2-2*c)^2)-(-k*phi(xi)^2+c*k+Omega)*(k*phi(xi)^2-Omega)*(phi(xi)^2-c)/(phi(xi)^2*(phi(xi)^2-2*c))-(1/2)*k^2*phi(xi)^2+Omega*k+(diff(phi(xi), xi))^2+1)*phi(xi) = 0

 (2)
> 

r2 := simplify(%)

(-4*phi(xi)^3*(-(1/2)*phi(xi)^2+c)^2*(diff(diff(phi(xi), xi), xi))+(-2*phi(xi)^6+4*phi(xi)^4*c)*(diff(phi(xi), xi))^2-2*phi(xi)^6+(-c^2*k^2+2*Omega*c*k-Omega^2+4*c)*phi(xi)^4+Omega^2*c^2)/(-2*phi(xi)^5+4*c*phi(xi)^3) = 0

 (3)
> 

pa := {diff(diff(phi(xi), xi), xi) = diff(psi(xi), xi), diff(phi(xi), xi) = psi(xi)}

{diff(diff(phi(xi), xi), xi) = diff(psi(xi), xi), diff(phi(xi), xi) = psi(xi)}

 (4)
> 

r3 := subs(pa, r2)

(-4*phi(xi)^3*(-(1/2)*phi(xi)^2+c)^2*(diff(psi(xi), xi))+(-2*phi(xi)^6+4*phi(xi)^4*c)*psi(xi)^2-2*phi(xi)^6+(-c^2*k^2+2*Omega*c*k-Omega^2+4*c)*phi(xi)^4+Omega^2*c^2)/(-2*phi(xi)^5+4*c*phi(xi)^3) = 0

 (5)
> 

psixif := isolate(r3, diff(psi(xi), xi))

diff(psi(xi), xi) = -(1/4)*(-(-2*phi(xi)^6+4*phi(xi)^4*c)*psi(xi)^2+2*phi(xi)^6-(-c^2*k^2+2*Omega*c*k-Omega^2+4*c)*phi(xi)^4-Omega^2*c^2)/(phi(xi)^3*(-(1/2)*phi(xi)^2+c)^2)

 (6)

NULL

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How to Separate Real and Imaginary Parts...

FZ 30 Maple 2022
complex real differential-equations
February 02 2025
3 2

I am trying to separate the real and imaginary parts of a complex expression in Maple to get Eq. (1.5) as in the attached image, but the Re and Im functions do not seem to return the expected results. Instead, Maple leaves the expression unchanged. PD_OD.mw

How to simplify Eqs. (3), (5) and (6) an...

FZ 30 Maple 2022
differentiation substitution pde
January 31 2025
0 5
> 

restart; with(PDEtools); declare(F(x, t), G(x, t), H(x, t))

F(x, t)*`will now be displayed as`*F

  

G(x, t)*`will now be displayed as`*G

  

H(x, t)*`will now be displayed as`*H

 (1)
> 

q := 1-(diff(diff(log(F(x, t)), x), t)); r := G/F; s := H/F

1-(diff(diff(F(x, t), t), x))/F(x, t)+(diff(F(x, t), x))*(diff(F(x, t), t))/F(x, t)^2

  

G/F

  

H/F

 (2)
> 

r1s1 := r*s; r1s1der := diff(r1s1(x, t), x)
> 

qt := diff(q(x, t), t)
> 

eq1B := F(x, t)^3*(qt+r1s1der) = 0; eq12B := simplify(expand(eq1B))

-F(x, t)^3*(diff((diff(diff(F(x, t), t), x))(x, t), t))/(F(x, t))(x, t)+F(x, t)^3*(diff(diff(F(x, t), t), x))(x, t)*(diff((F(x, t))(x, t), t))/(F(x, t))(x, t)^2+F(x, t)^3*(diff((diff(F(x, t), x))(x, t), t))*(diff(F(x, t), t))(x, t)/(F(x, t))(x, t)^2-2*F(x, t)^3*(diff(F(x, t), x))(x, t)*(diff(F(x, t), t))(x, t)*(diff((F(x, t))(x, t), t))/(F(x, t))(x, t)^3+F(x, t)^3*(diff(F(x, t), x))(x, t)*(diff((diff(F(x, t), t))(x, t), t))/(F(x, t))(x, t)^2+F(x, t)*(diff(G(x, t), x))*H(x, t)-2*G(x, t)*H(x, t)*(diff(F(x, t), x))+F(x, t)*G(x, t)*(diff(H(x, t), x)) = 0

 (3)
> 

D_x_x_G_F := (diff(G(x, t), x, x))*F(x, t)-2*(diff(G(x, t), x))*(diff(F(x, t), x))+G(x, t)*(diff(F(x, t), x, x)); D_t_t_F_F := F(x, t)*(diff(F(x, t), `$`(t, 2)))-2*(diff(F(x, t), t))^2

(diff(diff(G(x, t), x), x))*F(x, t)-2*(diff(G(x, t), x))*(diff(F(x, t), x))+G(x, t)*(diff(diff(F(x, t), x), x))

  

F(x, t)*(diff(diff(F(x, t), t), t))-2*(diff(F(x, t), t))^2

 (4)
> 

NULL
> 

rxt := diff(diff(r(x, t), x), t)
> 

eq2B := -2*q*r+rxt = 0
> 

eq22B := simplify(expand(eq2B))

((-F*F(x, t)*G(x, t)+2*G*F(x, t)^2)*(diff(diff(F(x, t), t), x))+(diff(diff(G(x, t), t), x))*F*F(x, t)^2+((2*F*G(x, t)-2*G*F(x, t))*(diff(F(x, t), x))-F*(diff(G(x, t), x))*F(x, t))*(diff(F(x, t), t))-(diff(G(x, t), t))*(diff(F(x, t), x))*F*F(x, t)-2*G*F(x, t)^3)/(F*F(x, t)^3) = 0

 (5)
> 

> 

sxt := diff(diff(s(x, t), x), t)
> 

eq3B := -2*q*s+sxt = 0
> 

eq32B := simplify(expand(eq3B))

((-F*F(x, t)*H(x, t)+2*H*F(x, t)^2)*(diff(diff(F(x, t), t), x))+(diff(diff(H(x, t), t), x))*F*F(x, t)^2+((2*F*H(x, t)-2*H*F(x, t))*(diff(F(x, t), x))-F*(diff(H(x, t), x))*F(x, t))*(diff(F(x, t), t))-(diff(H(x, t), t))*(diff(F(x, t), x))*F*F(x, t)-2*H*F(x, t)^3)/(F*F(x, t)^3) = 0

 (6)
> 

"#`# How to simplify Eqs. (3), (5) and (6) and write in terms of following bilineat operators` by using (4)"?""
> 

NULL

NULL

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