Lecture - Analysis MT22, XIII
Flashcards
What does it mean for $\sum^\infty _ {k = 0} c _ k x^k$ to be a real power series?
\[c_k, x \in \mathbb{R}\]
What does it mean for $\sum^\infty _ {k = 0} c _ k z^k$ to be a complex power series?
\[c_k, z \in \mathbb{C}\]
When considering power series, what is $0^0$ defined to be, by convention?
\[1\]
How is the exponential function $e^z = \exp(z)$ defined in terms of a power series?
\[\sum^\infty_{k = 0} \frac{z^k}{k!}\]
What’s a common technique for determining the radius of convergence of a power series?
The ratio test.
How is the sine function $\sin z$ defined in terms of a power series?
\[\sin z = \sum_{k = 0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)!}\]
How is the cosine function $\cos z$ defined in terms of a power series?
\[\cos z = \sum^\infty_{k = 0} \frac{(-1)^k z^{2k}}{(2k)!}\]
How is the hyperbolic sine function $\sinh z$ defined in terms of a power series?
\[\sinh z = \sum^\infty_{k = 0} \frac{z^{2k+1}}{(2k+1)!}\]
How is the hyperbolic cosine function $\cosh z$ defined in terms of a power series?
\[\cosh z = \sum^\infty_{k = 0} \frac{z^{2k}}{(2k)!}\]