This is basically a physics problem but I will try my best to highlight the mathematics behind it. Suppose I have two functions:

$$T(z,B)=\frac{\text{z}^3 e^{-3 A(\text{z})-B^2 \text{z}^2}}{4 \pi \int_0^{\text{z}} \xi ^3 e^{-3 A(\xi )-B^2 \xi ^2} \, d\xi },$$ $$\phi(z,B)=\int_0^z \sqrt{-\frac{2 \left(3 x A''(x)-3 x A'(x)^2+6 A'(x)+2 B^4 x^3+2 B^2 x\right)}{x}} \, dx$$ where $z \in \mathbb{R^+} $ and $B \in [0,1]$

and I want to find a function $A(z)$, which is known as the form factor in literature, such that the plot of the function $T(z,B)$ v/s $z$ has one minima along with the condition that $T(z,B)\rightarrow\infty$ when $z\rightarrow0$, also $\phi(z)$ is real valued. When I take the ansatz $A(z)=-a z^2$, I am able to satisfy the above condition for $B\in[0,0.6]$ and get plots like:

T1 vs z1

Now for a different model, I need to use such an ansatz for $A(z)$ such that I may be able to satisfy the real valued-ness of $\phi(z)$ and get minima as well as a maxima for the plot of $T(z,B)$ v/s $z$ with the condition that $T(z,B)\rightarrow\infty$ when $z\rightarrow0$ and $T(z,B)\rightarrow0$ when $z\rightarrow\infty$ to get plots like these:

T2 vs z2

If possible I want to keep $B\in[0,1]$. The constant $a$ has to be used in the form factor somehow, whose value is 0.15. The form factor can also be written in terms of $A(z,B)$. Is there a way to use Mathematical analysis to come up with such a form factor? Any help in this regard would be truly beneficial!



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