Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems Complex Analysis: Theory and Applications · Problem 7.2.6
Taylor and Laurent Series — Howell & Matthews
Mathematics — Complex Analysis · taylor-series-binomial
Problem Let f ( z ) = ( 1 + z ) β = exp [ β L o g ( 1 + z ) ] f(z) = (1+z)^{\beta} = \exp[\beta\,\mathrm{Log}(1+z)] f ( z ) = ( 1 + z ) β = exp [ β Log ( 1 + z )] ( 1 + z ) β (1+z)^{\beta} ( 1 + z ) β β \beta β z ∈ D 1 ( 0 ) z \in D_1(0) z ∈ D 1 ( 0 ) ( 1 + z ) β = 1 + β z + β ( β − 1 ) 2 ! z 2 + β ( β − 1 ) ( β − 2 ) 3 ! z 3 + ⋯ = 1 + ∑ n = 1 ∞ β ( β − 1 ) ( β − 2 ) ⋯ ( β − n + 1 ) n ! z n . \begin{aligned}
(1+z)^{\beta} &= 1 + \beta z + \frac{\beta(\beta-1)}{2!}z^2 + \frac{\beta(\beta-1)(\beta-2)}{3!}z^3 + \cdots \\
&= 1 + \sum_{n=1}^{\infty}\frac{\beta(\beta-1)(\beta-2)\cdots(\beta-n+1)}{n!}z^n.
\end{aligned} ( 1 + z ) β = 1 + β z + 2 ! β ( β − 1 ) z 2 + 3 ! β ( β − 1 ) ( β − 2 ) z 3 + ⋯ = 1 + n = 1 ∑ ∞ n ! β ( β − 1 ) ( β − 2 ) ⋯ ( β − n + 1 ) z n . Browse the problem library