Let $F:\mathbb{C}\rightarrow \mathbb{R}$ be a smooth function and $T$ be a positive real number.
Functions $f_{1},f_{2}:\mathbb{R}\rightarrow\mathbb{R}$ are differentiable.
Let $S[f_{1},f_{2}]:=\int_{0}^{T}F(f_1(t)+if_2(t))dt$
Now, consider such functions $f_1, f_2$ so that expression $S[f_1,f_2]$ is as low as it is possible.
Are there a system of differential equations, that is satisfied by $f_1,f_2$?
Is there an analogy to the Euler-Lagrange equation?
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