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finite_exponential_series [2017/03/18 17:30]
nikolaj created
finite_exponential_series [2017/03/18 17:42] (current)
nikolaj
Line 15: Line 15:
  
 $\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_n(z) = \mathrm{exp}_{n-1}(z) = \mathrm{exp}_n(z) - \dfrac{1}{n!} z^n$ $\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_n(z) = \mathrm{exp}_{n-1}(z) = \mathrm{exp}_n(z) - \dfrac{1}{n!} z^n$
 +
 +== Alternative ==
 +Another series which has $a^z$ as limit is the one given below. It has the nice feature that for integers $n<m$, it evaluates to $p(m,​n)=a^m$ exactly.
 +
 +<​code>​
 +p[m_, z_] := Sum[(a - 1)^k/k! \!\(
 +\*UnderoverscriptBox[\(\[Product]\),​ \(j = 0\), \(k - 1\)]\((z - 
 +       ​j)\)\),​ {k, 0, m}] // Expand
 +
 +Table[Table[p[m,​ z], {z, 1, m}], {m, 1, 7}]     // ​
 +  Simplify // TableForm
 +
 +i = 3;
 +2^i // N
 +p[5, i] /. {a -> 2} // N
 +
 +i = 3 + 1/2;
 +2^i // N
 +p[5, i] /. {a -> 2} // N
 +</​code>​
  
 === References === === References ===
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