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The Road to Reality Study Notes: Difference between revisions
→5.3 Multiple valuedness, natural logarithms
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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, $$z$$. | ||
Penrose goes further in representing $$z$$ with polar coordinates showing <math>z=logr+iθ</math>, then <math>e^z=re | Penrose goes further in representing $$z$$ with polar coordinates showing <math>z=logr+iθ</math>, then <math>e^z=re<sup>iθ</sup></math>. This formulation shows us that when we multiply two complex numbers, we take the product of their moduli and the sum of their arguments (using the addition to multiplication formula introduced in 5.2). | ||
Rounding out the chapter, Penrose gives us another further representation of assuming $$r=1$$, such that we recover the ‘unit circle’ in the complex plane with $$w=e^iθ$$. We can 'encapsulate the essentials of trigonometry in the much simpler properties of complex exponential functions' on this circle by showing <math>e^iθ=cos(θ) + isin(θ)</math>. | Rounding out the chapter, Penrose gives us another further representation of assuming $$r=1$$, such that we recover the ‘unit circle’ in the complex plane with $$w=e^iθ$$. We can 'encapsulate the essentials of trigonometry in the much simpler properties of complex exponential functions' on this circle by showing <math>e^iθ=cos(θ) + isin(θ)</math>. | ||