Difference between revisions of "Limit of Hilger real and imag parts yields classical"
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− | + | ==Theorem== | |
− | + | The following formula holds: | |
$$\displaystyle\lim_{h \rightarrow 0^+} \left[ \mathrm{Re}_h(z) + \mathring{\iota} \mathrm{Im}_h(z) \right] = \mathrm{Re}(z) + i \mathrm{Im}(z),$$ | $$\displaystyle\lim_{h \rightarrow 0^+} \left[ \mathrm{Re}_h(z) + \mathring{\iota} \mathrm{Im}_h(z) \right] = \mathrm{Re}(z) + i \mathrm{Im}(z),$$ | ||
where $\mathrm{Re}_h$ denotes the [[Hilger real part]] of $z$ and $\mathrm{Im}_h$ denotes the [[Hilger imaginary part]] of $z$. | where $\mathrm{Re}_h$ denotes the [[Hilger real part]] of $z$ and $\mathrm{Im}_h$ denotes the [[Hilger imaginary part]] of $z$. | ||
− | + | ||
− | + | ==Proof== | |
− | + | ||
− | + | ==References== | |
+ | |||
+ | [[Category:Theorem]] | ||
+ | [[Category:Unproven]] |
Latest revision as of 12:57, 17 August 2017
Theorem
The following formula holds: $$\displaystyle\lim_{h \rightarrow 0^+} \left[ \mathrm{Re}_h(z) + \mathring{\iota} \mathrm{Im}_h(z) \right] = \mathrm{Re}(z) + i \mathrm{Im}(z),$$ where $\mathrm{Re}_h$ denotes the Hilger real part of $z$ and $\mathrm{Im}_h$ denotes the Hilger imaginary part of $z$.