substituting functions in a function

2023 8 13
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I need to substitue funstions g(y) = (Wg/y)^2, h(x) = sqrt(1+g(sqrt(2)*x))+1/(sqrt(1+g(sqrt(2)*x))-1), V(z) = z-1/(z+1) and f(w) = sqrt(sqrt(w)+1/(sqrt(w)-1)) in final equation theta = ln(V[f(g(sqrt2)+1)]/V[f(g(Lg))]) + ln(h(2.3*t)/h(s)). I am using nested equation but not able to solve it. Please help.

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Walter Roberson
Walter Roberson 2023-8-13
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Ran in:
You will need to check that I transcribed everything correctly.
syms Lg Wg s t w x y z
g(y) = (Wg/y)^2
g(y) = 
Root2 = sqrt(sym(2));
h(x) = sqrt(1+g(Root2*x))+1/(sqrt(1+g(Root2*x))-1)
h(x) = 
V(z) = z-1/(z+1)
V(z) = 
f(w) = sqrt(sqrt(w)+1/(sqrt(w)-1))
f(w) = 
Theta = log(V(f(g(Root2)+1))/V(f(g(Lg)))) + log(h(sym(2.3)*t)/h(s))
Theta = 
simplify(Theta, 'steps', 50)
ans = 

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Marsha Parmar
Marsha Parmar 2023-8-14
编辑：Marsha Parmar 2023-8-14
Thank you for the answer. But may be I was not able to explain my self clearly. There is a big theta equation which goes as,
θ_total=1/(πWg k) ln(V[f(g[√2 s])+1]/V[f(g[Lg ])] )+1/2πsk ln((h(2.3t))/(h(s))).
Here you can see that 'V' is function of 'f' which in turn is a function of 'g' etc. Now these functions are given as,
g(y)=(Wg/y)^2
f(w)= √((√w+1)/(√w-1) )
V(z)=(z-1)/(z+1)
h(x)= (√(1+g(√2 x))+1)/(√(1+g(√2 x))-1).
So now suppose I wish to substiute g in h(x), then it will be,
h(x)= (√(1+(Wg/(√2 x))^2 )+1)/(√(1+(Wg/(√2 x))^2 )-1)
which I have done manually. Now if I need to calculate,
V[f(g[√2 s]), where 'g' is function of √2 s, 'f' is a function of f(g[√2 s]), and then 'V' is afunction of V[f(g[√2 s]).
If there is a way to write down a cose to substitute these function into a bigger function it will be very helpful.
Torsten
Torsten 2023-8-14
But that's exactly what @Walter Roberson 's code does.
It defines functions g(y), f(w), V(z) and h(x) and computes a new function Theta by certain compositions of the functions previously defined.
Marsha Parmar
Marsha Parmar 2023-8-14
Yes, I got to see now. Thank you so much.

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