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Consider the following function $$f(x) = \left\{\begin{array}-(a^2 - x^2)^{1/2} & \text{if }|x|\leq a \\+ \infty &\text{otherwise}\end{array}\right.$$ and compute its Legendre transform.

I solved the above questions, however, I can't solve the question that the infinite in the function. How can I apply the Legendre transform to this question?

The Amplitwist
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Lotus72
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    Hint : The legendre transform is defined by $f^(x^) = \sup_{x\in \mathbb R}xx^* - f(x)$. Since $-f(x) = -\infty$ for $x\notin[-a,a]$ you have : $$f^(x^) = \sup_{x\in[-a,a]} xx^* - f(x)$$ From which point you can do the usual calculations to find $f^*$ – SolubleFish Jun 05 '21 at 18:52

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