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Is there a name for the set of complex numbers over affinely extended real line, that is $\mathbb{C}\cup \{-\infty\}\cup\{+\infty\}$? I think this set is the most commonly used in analysis applications because the both extensions (complex numbers and affine reals) proved to be hugely fruitful in analysis.

For instance, this is a formula for inverse Mellin's transform from Wikipedia: $$\left\{\mathcal{M}^{-1}\varphi\right\}(x) = f(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} \varphi(s)\, ds$$

But I never encountered a name for this widely used numerical system.

Anixx
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    Is it widely used? I have never seen it used at all, in fact. – Mariano Suárez-Álvarez Aug 23 '15 at 08:49
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    @Mariano Suárez-Alvarez Look for instance at the Wiki article about Forier transform. It uses both complex numbers and signed infinities a lot: https://en.wikipedia.org/wiki/Fourier_transform – Anixx Aug 23 '15 at 08:54
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    @Mariano Suárez-Alvarez And inverse Mellin tranform uses the both in one formula: https://upload.wikimedia.org/math/5/4/a/54aca715828a45c0f8e1c3514fe26dcf.png – Anixx Aug 23 '15 at 08:55
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    That notation has nothing to to with extended anything, really. – Mariano Suárez-Álvarez Aug 23 '15 at 09:01
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    Note that your example is more like viewing $\mathbf{C} \subseteq \bar{\mathbf{R}} \times \bar{\mathbf{R}}$ rather than adding two extra points to the complex numbers. (where $\bar{\mathbf{R}}$ is the extended real numbers) –  Aug 23 '15 at 10:10
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    @Hurkyl agreed, but what's the usual name (or symbol) for such set? – Anixx Aug 23 '15 at 10:14

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