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The Road to Reality Study Notes: Difference between revisions
→5.3 Multiple valuedness, natural logarithms
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=== 5.3 Multiple valuedness, natural logarithms === | === 5.3 Multiple valuedness, natural logarithms === | ||
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 | 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+\frac{1}{1!}+\frac{1}{2!}+\frac{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 with $$z+2πin$$, where $$n$$ is any integer we care to choose. This represents a full rotation of $$2π$$ in the complex plane with all multiples of $$n$$ achieving the same point, $$z$$. | 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 with $$z+2πin$$, where $$n$$ is any integer we care to choose. This represents a full rotation of $$2π$$ in the complex plane with all multiples of $$n$$ achieving the same point, $$z$$. | ||