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

The Road to Reality Study Notes (view source)

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 === 5.4 Complex Powers ===
 Returning to the ambiguity problem of multi-valuedness, it seems the best way to avoid issues is when a particular choice of $$logw$$ has been specified.  As an example, $$w^z$$ with $$z=1/2$$.  We can specify a rotation for $$logw$$ to achieve $$+w^1/2$$, then another rotation of $$logw$$ to achieve $$-w^1/2$$.  The sign change is achieved because of the Euler formula <math>e^{π*i}=-1</math>. Note the process:
-<math>w^z=e^{zlogw}=e^{zre^{iθ}}=e^{ze^iθ}}</math>, then specifying rotations for theta allows us to achieve either $$+w^1/2$$ or $$-w^1/2$$.
+<math>w^z=e^{zlogw}=e^{zre^{iθ}}=e^{ze^{iθ}}</math>, then specifying rotations for theta allows us to achieve either $$+w^1/2$$ or $$-w^1/2$$.
 Penrose notes an interesting curiosity for the quantity $$i^i$$.  We can specify <math>logi=(1/2)πi</math> because of the general relationship <math>logw=logr+iθ</math>.  If $$w=i$$, then its easy to see <math>logi=(1/2)πi</math> from noting that y is on the vertical axis in the complex plane (rotation of $$π/2$$).  This specification, and all rotations, amazingly achieve real number values for i^i.