Difference between revisions of "The Road to Reality Study Notes"

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→‎5.3 Multiple valuedness, natural logarithms

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 === 5.3 Multiple valuedness, natural logarithms ===
-We need to be careful with the above assertion of the logarithm, mainly since $$b^z$$ and logbw are ‘many valued’.  Solving the equations would require a particular choice for b to isolate the solution.  With this, the ‘base of natural logarithms’ is introduced as the number ‘e’, whose definition is the power series 1+1/1!+1/2!+1/3!+…If b=e, then b^z=e^z=1+z/1!+z^2/2!+z^3/3!+…This power series converges for all values of z which then makes for an interesting choice to solve the ambiguity problem above.  Thus we can rephrase the problem above with the natural logarithm, z=logw if w=e^z.
+We need to be careful with the above assertion of the logarithm, mainly since $$b^z$$ and <math>log_bw</math> are ‘many valued’.  Solving the equations would require a particular choice for $$b$$ to isolate the solution.  With this, the ‘base of natural logarithms’ is introduced as the [https://en.wikipedia.org/wiki/E_(mathematical_constant) number e], whose definition is the power series <math>1+1/1!+1/2!+1/3!+…</math>. This power series converges for all values of z which then makes for an interesting choice to solve the ambiguity problem above.  Thus we can rephrase the problem above with the natural logarithm, <math>z=logw</math> if $$w=e^z$$.
 However, even with this natural logarithm we run into multi-valuedness ambiguity from above.  Namely that z still has many values that lead to the same solution, z+2*pi*i*n, where n is any integer we care to choose.  This represents a full rotation of 2*pi in the complex plane which achieves the same point, z.