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Question: How can I=sqrt(-1) can be properly factored out from the square roots?

Question:How can I=sqrt(-1) can be properly factored out from the square roots?

Ahmed111 135 Maple 2025
radical simplify simplification edited-question
4 hours ago
0

I am trying to factor out I = sqrt(-1) from square roots in my Maple expression by using a substitution f2. However, after applying these substitutions to my final expression, there is no visible change. In addition, the term sqrt(2)/2 + sqrt(2)*I/2 also appear. How can I=sqrt(-1) can be properly factored out from the square roots?

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restart
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with(Student[Precalculus])
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interface(showassumed = 0)
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assume(x::real); assume(t::real); assume(lambda1::complex); assume(lambda2::complex); assume(a::real); assume(A__c::real); assume(B1::real); assume(B2::real); assume(delta1::real); assume(delta2::real); assume(`ω__0`::real); assume(g::real); assume(l__0::real)
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expr := (0*A__c)*exp(-(2*I)*(A__c^2*g*l__0^2-1/2)*`ω__0`*t)+(2*I)*exp(-I*(A__c^2*g*l__0^2-1/2)*`ω__0`*t)*(sqrt(delta1+I*delta2-sqrt(-A__c^2*g+(delta1+I*delta2)^2))*exp(-2*sqrt(-A__c^2*g+(delta1+I*delta2)^2)*(l__0^2*(I*delta1-delta2)*t*`ω__0`+(1/2)*x))-sqrt(delta1+I*delta2+sqrt(-A__c^2*g+(delta1+I*delta2)^2))*exp(sqrt(-A__c^2*g+(delta1+I*delta2)^2)*(x+(2*I)*`ω__0`*l__0^2*(delta1+I*delta2)*t)))*(sqrt(-delta1+I*delta2-sqrt(-A__c^2*g+(delta1-I*delta2)^2))*exp((2*(l__0^2*(I*delta1+delta2)*t*`ω__0`-(1/2)*x))*sqrt(-A__c^2*g+(delta1-I*delta2)^2))-sqrt(-delta1+I*delta2+sqrt(-A__c^2*g+(delta1-I*delta2)^2))*exp(-(2*(l__0^2*(I*delta1+delta2)*t*`ω__0`-(1/2)*x))*sqrt(-A__c^2*g+(delta1-I*delta2)^2)))*delta2/(exp(I*(A__c^2*g*l__0^2-1/2)*`ω__0`*t)*(((-sqrt(delta1+I*delta2-sqrt(-A__c^2*g+(delta1+I*delta2)^2))*sqrt(-delta1+I*delta2+sqrt(-A__c^2*g+(delta1-I*delta2)^2))-sqrt(delta1+I*delta2+sqrt(-A__c^2*g+(delta1+I*delta2)^2))*sqrt(-delta1+I*delta2-sqrt(-A__c^2*g+(delta1-I*delta2)^2)))*exp((2*(l__0^2*(I*delta1+delta2)*t*`ω__0`-(1/2)*x))*sqrt(-A__c^2*g+(delta1-I*delta2)^2))+exp(-(2*(l__0^2*(I*delta1+delta2)*t*`ω__0`-(1/2)*x))*sqrt(-A__c^2*g+(delta1-I*delta2)^2))*(sqrt(delta1+I*delta2-sqrt(-A__c^2*g+(delta1+I*delta2)^2))*sqrt(-delta1+I*delta2-sqrt(-A__c^2*g+(delta1-I*delta2)^2))+sqrt(-delta1+I*delta2+sqrt(-A__c^2*g+(delta1-I*delta2)^2))*sqrt(delta1+I*delta2+sqrt(-A__c^2*g+(delta1+I*delta2)^2))))*exp(-2*sqrt(-A__c^2*g+(delta1+I*delta2)^2)*(l__0^2*(I*delta1-delta2)*t*`ω__0`+(1/2)*x))+exp(sqrt(-A__c^2*g+(delta1+I*delta2)^2)*(x+(2*I)*`ω__0`*l__0^2*(delta1+I*delta2)*t))*((sqrt(delta1+I*delta2-sqrt(-A__c^2*g+(delta1+I*delta2)^2))*sqrt(-delta1+I*delta2-sqrt(-A__c^2*g+(delta1-I*delta2)^2))+sqrt(-delta1+I*delta2+sqrt(-A__c^2*g+(delta1-I*delta2)^2))*sqrt(delta1+I*delta2+sqrt(-A__c^2*g+(delta1+I*delta2)^2)))*exp((2*(l__0^2*(I*delta1+delta2)*t*`ω__0`-(1/2)*x))*sqrt(-A__c^2*g+(delta1-I*delta2)^2))-exp(-(2*(l__0^2*(I*delta1+delta2)*t*`ω__0`-(1/2)*x))*sqrt(-A__c^2*g+(delta1-I*delta2)^2))*(sqrt(delta1+I*delta2-sqrt(-A__c^2*g+(delta1+I*delta2)^2))*sqrt(-delta1+I*delta2+sqrt(-A__c^2*g+(delta1-I*delta2)^2))+sqrt(delta1+I*delta2+sqrt(-A__c^2*g+(delta1+I*delta2)^2))*sqrt(-delta1+I*delta2-sqrt(-A__c^2*g+(delta1-I*delta2)^2)))))*(-delta1+I*delta2)*(delta1+I*delta2))

(2*I)*exp(-I*(A__c^2*g*l__0^2-1/2)*omega__0*t)*((delta1+I*delta2-(-A__c^2*g+(delta1+I*delta2)^2)^(1/2))^(1/2)*exp(-2*(-A__c^2*g+(delta1+I*delta2)^2)^(1/2)*(l__0^2*(I*delta1-delta2)*t*omega__0+(1/2)*x))-(delta1+I*delta2+(-A__c^2*g+(delta1+I*delta2)^2)^(1/2))^(1/2)*exp((-A__c^2*g+(delta1+I*delta2)^2)^(1/2)*(x+(2*I)*omega__0*l__0^2*(delta1+I*delta2)*t)))*((-delta1+I*delta2-(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))^(1/2)*exp(2*(l__0^2*(I*delta1+delta2)*t*omega__0-(1/2)*x)*(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))-(-delta1+I*delta2+(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))^(1/2)*exp(-2*(l__0^2*(I*delta1+delta2)*t*omega__0-(1/2)*x)*(-A__c^2*g+(delta1-I*delta2)^2)^(1/2)))*delta2/(exp(I*(A__c^2*g*l__0^2-1/2)*omega__0*t)*(((-(delta1+I*delta2-(-A__c^2*g+(delta1+I*delta2)^2)^(1/2))^(1/2)*(-delta1+I*delta2+(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))^(1/2)-(delta1+I*delta2+(-A__c^2*g+(delta1+I*delta2)^2)^(1/2))^(1/2)*(-delta1+I*delta2-(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))^(1/2))*exp(2*(l__0^2*(I*delta1+delta2)*t*omega__0-(1/2)*x)*(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))+exp(-2*(l__0^2*(I*delta1+delta2)*t*omega__0-(1/2)*x)*(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))*((delta1+I*delta2-(-A__c^2*g+(delta1+I*delta2)^2)^(1/2))^(1/2)*(-delta1+I*delta2-(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))^(1/2)+(-delta1+I*delta2+(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))^(1/2)*(delta1+I*delta2+(-A__c^2*g+(delta1+I*delta2)^2)^(1/2))^(1/2)))*exp(-2*(-A__c^2*g+(delta1+I*delta2)^2)^(1/2)*(l__0^2*(I*delta1-delta2)*t*omega__0+(1/2)*x))+exp((-A__c^2*g+(delta1+I*delta2)^2)^(1/2)*(x+(2*I)*omega__0*l__0^2*(delta1+I*delta2)*t))*(((delta1+I*delta2-(-A__c^2*g+(delta1+I*delta2)^2)^(1/2))^(1/2)*(-delta1+I*delta2-(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))^(1/2)+(-delta1+I*delta2+(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))^(1/2)*(delta1+I*delta2+(-A__c^2*g+(delta1+I*delta2)^2)^(1/2))^(1/2))*exp(2*(l__0^2*(I*delta1+delta2)*t*omega__0-(1/2)*x)*(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))-exp(-2*(l__0^2*(I*delta1+delta2)*t*omega__0-(1/2)*x)*(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))*((delta1+I*delta2-(-A__c^2*g+(delta1+I*delta2)^2)^(1/2))^(1/2)*(-delta1+I*delta2+(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))^(1/2)+(delta1+I*delta2+(-A__c^2*g+(delta1+I*delta2)^2)^(1/2))^(1/2)*(-delta1+I*delta2-(-A__c^2*g+(delta1-I*delta2)^2)^(1/2))^(1/2))))*(I*delta2-delta1)*(delta1+I*delta2))

 (1)
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`assuming`([simplify(combine(simplify(convert(combine(eval(expr, delta1 = 0)), trigh))))], [delta2 > g*A__c and g*A__c > 0])

(cos((2*A__c^2*g*l__0^2-1)*omega__0*t)-I*sin((2*A__c^2*g*l__0^2-1)*omega__0*t))*(-I*cosh(4*(l__0^2*delta2*t*omega__0-(1/2)*x)*(-A__c^2*g-delta2^2)^(1/2))*delta2+(I*delta2-(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(I*delta2+(-A__c^2*g-delta2^2)^(1/2))^(1/2)+sinh(4*(l__0^2*delta2*t*omega__0-(1/2)*x)*(-A__c^2*g-delta2^2)^(1/2))*(-A__c^2*g-delta2^2)^(1/2))/(delta2*(I*(I*delta2-(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(I*delta2+(-A__c^2*g-delta2^2)^(1/2))^(1/2)*cosh(4*(l__0^2*delta2*t*omega__0-(1/2)*x)*(-A__c^2*g-delta2^2)^(1/2))+delta2))

 (2)
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f1 := simplify(convert(numer(%),exp))/factor(denom(%))

I*exp(-(2*I)*(A__c^2*g*l__0^2-1/2)*omega__0*t)*(-I*cosh(4*(l__0^2*delta2*t*omega__0-(1/2)*x)*(-A__c^2*g-delta2^2)^(1/2))*delta2+(I*delta2-(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(I*delta2+(-A__c^2*g-delta2^2)^(1/2))^(1/2)+sinh(4*(l__0^2*delta2*t*omega__0-(1/2)*x)*(-A__c^2*g-delta2^2)^(1/2))*(-A__c^2*g-delta2^2)^(1/2))/((-(I*delta2-(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(I*delta2+(-A__c^2*g-delta2^2)^(1/2))^(1/2)*cosh(2*(2*delta2*l__0^2*t*omega__0-x)*(-A__c^2*g-delta2^2)^(1/2))+I*delta2)*delta2)

 (3)
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sqrtterms := indets(%, sqrt)

{(I*delta2-(-A__c^2*g-delta2^2)^(1/2))^(1/2), (I*delta2+(-A__c^2*g-delta2^2)^(1/2))^(1/2), (-A__c^2*g-delta2^2)^(1/2)}

 (4)
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f2 := subs({sqrtterms[1] = sqrt(I)*sqrt(delta2-sqrt(-A__c^2*g-delta2^2)/(I)), sqrtterms[2] = sqrt(I)*sqrt(delta2+sqrt(-A__c^2*g-delta2^2)/(I)), sqrtterms[3] = sqrt(I)*sqrt(A__c^2*g+delta2^2)})

{(I*delta2-(-A__c^2*g-delta2^2)^(1/2))^(1/2) = ((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2), (I*delta2+(-A__c^2*g-delta2^2)^(1/2))^(1/2) = ((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2), (-A__c^2*g-delta2^2)^(1/2) = ((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*(A__c^2*g+delta2^2)^(1/2)}

 (5)
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f3 := subs(f2, f1)

I*exp(-(2*I)*(A__c^2*g*l__0^2-1/2)*omega__0*t)*(-I*cosh(4*(l__0^2*delta2*t*omega__0-(1/2)*x)*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*(A__c^2*g+delta2^2)^(1/2))*delta2+((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))^2*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)+sinh(4*(l__0^2*delta2*t*omega__0-(1/2)*x)*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*(A__c^2*g+delta2^2)^(1/2))*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*(A__c^2*g+delta2^2)^(1/2))/((-((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))^2*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*cosh(2*(2*delta2*l__0^2*t*omega__0-x)*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*(A__c^2*g+delta2^2)^(1/2))+I*delta2)*delta2)

 (6)
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f4 := subs({sqrt(A__c^2*g+delta2^2) = Z}, f3)

I*exp(-(2*I)*(A__c^2*g*l__0^2-1/2)*omega__0*t)*(-I*cosh(4*(l__0^2*delta2*t*omega__0-(1/2)*x)*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*Z)*delta2+((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))^2*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)+sinh(4*(l__0^2*delta2*t*omega__0-(1/2)*x)*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*Z)*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*Z)/((-((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))^2*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*cosh(2*(2*delta2*l__0^2*t*omega__0-x)*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*Z)+I*delta2)*delta2)

 (7)
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f4f := A__c*exp(-(2*I)*(A__c^2*g*l__0^2-1/2)*`ω__0`*t)+f4

A__c*exp(-(2*I)*(A__c^2*g*l__0^2-1/2)*omega__0*t)+I*exp(-(2*I)*(A__c^2*g*l__0^2-1/2)*omega__0*t)*(-I*cosh(4*(l__0^2*delta2*t*omega__0-(1/2)*x)*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*Z)*delta2+((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))^2*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)+sinh(4*(l__0^2*delta2*t*omega__0-(1/2)*x)*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*Z)*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*Z)/((-((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))^2*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*cosh(2*(2*delta2*l__0^2*t*omega__0-x)*((1/2)*2^(1/2)+((1/2)*I)*2^(1/2))*Z)+I*delta2)*delta2)

 (8)
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f4fnl := subs({I = -I, x = -x}, f4f)

A__c*exp((2*I)*(A__c^2*g*l__0^2-1/2)*omega__0*t)-I*exp((2*I)*(A__c^2*g*l__0^2-1/2)*omega__0*t)*(I*cosh(4*(l__0^2*delta2*t*omega__0+(1/2)*x)*((1/2)*2^(1/2)-((1/2)*I)*2^(1/2))*Z)*delta2+((1/2)*2^(1/2)-((1/2)*I)*2^(1/2))^2*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)+sinh(4*(l__0^2*delta2*t*omega__0+(1/2)*x)*((1/2)*2^(1/2)-((1/2)*I)*2^(1/2))*Z)*((1/2)*2^(1/2)-((1/2)*I)*2^(1/2))*Z)/((-((1/2)*2^(1/2)-((1/2)*I)*2^(1/2))^2*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*cosh(2*(2*delta2*l__0^2*t*omega__0+x)*((1/2)*2^(1/2)-((1/2)*I)*2^(1/2))*Z)-I*delta2)*delta2)

 (9)
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Mdensity := simplify(f4f*f4fnl)

(1/4)*(2*(1-I*A__c*cosh((1-I)*(2*delta2*l__0^2*t*omega__0+x)*2^(1/2)*Z)*delta2)*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)-2*cosh((1-I)*(2*delta2*l__0^2*t*omega__0+x)*2^(1/2)*Z)*delta2+(1+I)*2^(1/2)*Z*sinh((1-I)*(2*delta2*l__0^2*t*omega__0+x)*2^(1/2)*Z)+(2*I)*A__c*delta2^2)*(2*(I*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*A__c-1)*delta2*cosh((1+I)*(2*delta2*l__0^2*t*omega__0-x)*2^(1/2)*Z)+(1-I)*2^(1/2)*Z*sinh((1+I)*(2*delta2*l__0^2*t*omega__0-x)*2^(1/2)*Z)-(2*I)*A__c*delta2^2+2*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2))/(delta2^2*((delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*cosh((1+I)*(2*delta2*l__0^2*t*omega__0-x)*2^(1/2)*Z)-delta2)*((delta2-I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*(delta2+I*(-A__c^2*g-delta2^2)^(1/2))^(1/2)*cosh((1-I)*(2*delta2*l__0^2*t*omega__0+x)*2^(1/2)*Z)-delta2))

 (10)

NULL

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