By Kerry Eggers, Dwight Jaynes
The path Blazers have risen to the head of the basketball global yet have come away simply in need of their target of an NBA championship. The publication explores why the path Blazers have arise brief and appears on the historical past that led them to the pinnacle in their
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Extra resources for Against the World: A Behind-The-Scenes Look at the Portland Trail Blazers' Chase for the Nba Championship
This complex function coincides with f (x) for z = x by definition and thereby provides a unique analytic continuation of f (x) due to the theorem, which states that if there were another analytic continuation it would have coincided with f (z) anyway. For√example, suppose we are given the following function of real variable f (x) = x − 1 defined on [1, ∞). We therefore√know that an analytic continuation of this function to the complex plane is f (z) = z − 1. Moreover, assuming the standard definition of the argument, we also know that this function has a branch cut on (−∞, 1], with √ the limiting values being f (x ± iδ) = ±i 1 − x.
Integrating n times by parts, we have: ↔ f (t) = t n F( p) = n! p n+1 . 41) Using a similar procedure one can evaluate the image of the derivative, ∞ dt f (t) e− pt = f (t)e− pt 0 ∞ 0 ∞ +p dt f (t) e− pt = p F( p) − f (0). 42) For the image of the n th order derivative then follows: f (n) (t) → p n F( p) − p n−1 f (0) − p n−2 f (0) − · · · − p f (n−2) (0) − f (n−1) (0). ,n−1) (0) = 0 then f (n) (t) → p n F( p) which is probably the most important property of the Laplace transform. Analogous relations can be obtained for the derivative of the image.
The Cauchy theorem states: ⎧ ⎪ ⎪ ⎨ = CR ± ⎪ ⎪ ⎩C ± + ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ C± + 2πi Res f (z). 3 Branch Cut Integration 33 On C ± our function has essentially the same arguments as before: f (x ± iδ) = e±iπa a 1−x 1+x R(x) , x − x0 which, in the limit, produces the principal value integral as = −2i sin(πa)Ia (x0 ). lim →0 ± C± On the other hand on C ± we put z = x0 + eiθ to recover in the limit → 0 (like in Sect. 3) the same arguments as on C ± , the θ–integration, being in the same direction (anti-clockwise), results in the factor iπ (half-residue) for each semi-circle: e±iπa = iπ ± C± ± 1 − x0 1 + x0 a 1 − x0 1 + x0 R(x0 ) = 2πi cos(πa) a R(x0 ).
Against the World: A Behind-The-Scenes Look at the Portland Trail Blazers' Chase for the Nba Championship by Kerry Eggers, Dwight Jaynes