Difference between revisions of "Abs(e^z-1) less than or equal to e^(abs(z))-1 less than or equal to abs(z)e^(abs(z))"
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(Created page with "==Theorem== The following formula holds for all $z \in \mathbb{C}$: $$\left| e^z - 1 \right| \leq e^{|z|}-1 \leq |z| e^{|z|},$$ where $e^z$ denotes the exponential. ==Proo...") |
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==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Abs(z)/4 less than abs(e^z-1) less than (7abs(z))/4 for 0 less than abs(z) less than 1|next=}}: 4.2.39 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Abs(z)/4 less than abs(e^z-1) less than (7abs(z))/4 for 0 less than abs(z) less than 1|next=findme}}: 4.2.39 |
[[Category:Theorem]] | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 00:39, 23 December 2016
Theorem
The following formula holds for all $z \in \mathbb{C}$: $$\left| e^z - 1 \right| \leq e^{|z|}-1 \leq |z| e^{|z|},$$ where $e^z$ denotes the exponential.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.2.39
