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We've come across exponential functions like 2^x ,3^x etc. Let me take one of them, say y=2^x here; if we consider y to be the length of the tree after X years, it has a meaningful explanation, that is, say X=2, then we can say that after 2 years the tree was 4 times its original length, its length increasing by a factor of 2 compared to previous year's growth. So what does a function like y=2^ixmean ? What does $i$ do here?

J. W. Tanner
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user794763
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    First, note that all exponents are the same, so really you can just focus on, say, $e^{ix}$. This sweeps out a circle in the complex plane. – Cameron Williams Sep 08 '20 at 12:36
  • $2^x$ has some power series $a_0+a_1x+\ldots$ Then we say let $2^{ix}=a_0+a_1(ix)+\ldots$ by analogy. – Chrystomath Sep 08 '20 at 13:07
  • $2^{ix}$ (for x real) is a complex number. Complex Analysis is the analytic continuation of Real Analysis. You would be amazed how real-world problems, analytically-continued into the complex domain are used to find real solutions. The most beautiful example of analytic continuation in mathematics I think is the Riemann zeta function used in a proof of the Prime Number Theorem dealing with real numbers. – Dominic Sep 08 '20 at 13:09
  • @Dominic could u elaborate the meaning of the equation alone coz my knowledge of reimann zeta function is limited to just numberphile videos – user794763 Sep 08 '20 at 13:17

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