Further Maths - Complex Sums
See Also
- [[Further Maths - Complex Numbers]]A
- [[Further Maths - Exponential Form of Complex Numbers]]A
- [[Further Maths - Trig Equations with Complex Numbers]]A
- [[Further Maths - Roots of Complex Numbers]]A
Flashcards
2022-03-30
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\[w + wz + wz^2 + \cdots + wz^{n-1}\]What is this equal to??
\[\frac{w(z^n - 1)}{z - 1}\]#####
\[\sum^{n - 1} _ {r = 0} wz^r\]What is this equal to??
\[\frac{w(z^n - 1)}{z - 1}\]#####
\[w + wz + wz^2 + \cdots + wz^{n}\]What is this equal to?? Remember the plus one! Don’t you dare skip this card if you didn’t say that explicitly!
\[\frac{w(z^{n+1} - 1)}{z - 1}\]#####
\[\sum^{n} _ {r = 0} wz^r\]What is this equal to?? Remember the plus one! Don’t you dare skip this card if you didn’t say that explicitly!
\[\frac{w(z^n - 1)}{z - 1}\]What is the point of the complex sums of series topic?
Turning complicated series into simple expressions of $\sin$ and $\cos$.
What are the two important results when working out complex geometric sums?
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\[\frac{-2}{e^{\frac{\pi i}{n}} - 1}\]What might your next step be here?? You’re trying to create the difference or sum of two opposite pairs, multiply the bottom by $e^{-\frac{\pi i}{2n}}$
\[-2\frac{e^{-\frac{\pi i}{2n}}}{e^{\pi i}{2n} - e^{-\frac{\pi i}{2n}}}\]What are the two techniques for making $z + 1/z$ or $z - 1/z$ terms appear from nothing in the complex sums topic?
- Factoring out
- Multiplying top and bottom
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\[1 + z + z^2 + \cdots + z^{2n-1}\]What is this equal to??
\[\frac{1(z^{2n} - 1)}{z - 1}\]#####
\[1 + z + z^2 + \cdots + z^n\]What is this equal to??
\[\frac{1(z^{n+1} - 1)}{z - 1}\]2022-04-14
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\[\frac{-2e^{-\frac{\pi}{2n}i}}{2i\sin\left( \frac{\pi}{2n} \right)}\]How could you simplify this?? Use the fact that $-2/2i = i$
\[\frac{ie^{-\frac{\pi}{2n}}{\sin\left( \frac{\pi}{2n} \right)}\]