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What are the branches of maths where we can see undoubtful connections with economics? Where can we use mathematical methods or models and apply them to analyze economic concepts?

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Not really a fully-fledged answer, but more of an interesting fact: John Maynard Keynes, arguably the most renowned Economist ever, was a Cambridge graduate in Mathematics. Also, Linear Regressions en.wikipedia.org/wiki/Linear_regression are used quite often in examining Economic relationships. – user64219 Sep 13 '12 at 10:17

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Lots of branches of maths are used in lots of areas of economics, so it would be hard to give a concise answer. Have a look at Essential Mathematics for Economic Analysis by Sydsaeter & Hammond, and the follow-up Further Mathematics for Economic Analysis.

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