I am currently studying Stephen Boyd, where the conjugate function is defined to be $f^*(y) = \underset{x \in Dom(f)}{sup} (y^Tx-f(x))$.
I understand the definition, but when I search for more resources online, I find something called Legendre transform, which is defined as $\displaystyle{f^{*}(x^{*}) = \sup _{x\in I}(x^{*}x-f(x)),\ \ \ \ x^{*}\in I^{*}}$, where $f$ is convex.
These definitions seem exactly the same, except for the condition on $f$.
Also, now I think about the conjugate of complex numbers and the conjugate of complex functions, which are defined to be the same quantity, but with an inverted sign of the imaginary part.
There are also conjugate pairs (Lagrangian and Hamiltonian) in physics, which rely on similar concepts.
I am guessing they are related closely. Could you help me understand these cohesively and what general conjugate functions are?
References: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
https://som-course.github.io/opt-fall-2021/static_files/Notes/lecture3.pdf