Zeration

[JmsNxn]

This definition for zeration has similar properties to the common zeration:

\( a \circ b \,=\, ln(e^a+e^b) \)


\( a \circ -\infty \,=\, a \,+\, ln(1),=\, a \)
\( a \circ a \,=\, a \,+\, ln(2) \)
\( (a_n \circ \,...\,(a_3 \circ (a_2 \circ a_1))) \,=\, a \,+\, ln(n) \)

if b>>a
\( a \circ b \,=\, b \,+\ ln(1+e^{a-b}) \,\approx \, b \)

if b>a
\( a \circ b \,=\, b \,+\ ln(1+e^{a-b}) \,<\, b \,+\, ln(2) \)


\( a \circ -\infty \,=\, -\infty \circ a \,=\, a \)

\( -\infty  \circ -\infty \,=\,-\infty \)


\( x  \,\circ\, -x \,=\, ln(cosh(x)) \,+\, ln(2) \)

Under that definition of zeration, a°-∞ would be ln(e^a+e^-∞). Not a.

Hey, Catullus. Those are one and the same on the extended real number system.

\[
\log(e^a + e^{-\infty}) = \log(e^a + 0) = \log(e^a) = a\\
\]