03/20/2015, 09:44 AM
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
\( a \circ b \,=\, b \,+\ ln(1+e^{a-b}) \,<\, b \,+\, ln(2) \)
\( a \circ -\infty \,=\, -\infty \circ a \,=\, a \)
\( -\infty \circ -\infty \,=\,-\infty \)
\( x \,\circ\, -x \,=\, ln(cosh(x)) \,+\, ln(2) \)
\( a \circ b \,=\, ln(e^a+e^b) \)
\( a \circ -\infty \,=\, a \,+\, ln(1),=\, a \)
\( a \circ a \,=\, a \,+\, ln(2) \)
\( (a_n \circ \,...\,(a_3 \circ (a_2 \circ a_1))) \,=\, a \,+\, ln(n) \)
if b>>a
\( a \circ b \,=\, b \,+\ ln(1+e^{a-b}) \,\approx \, b \)
if b>a
\( a \circ b \,=\, b \,+\ ln(1+e^{a-b}) \,<\, b \,+\, ln(2) \)
\( a \circ -\infty \,=\, -\infty \circ a \,=\, a \)
\( -\infty \circ -\infty \,=\,-\infty \)
\( x \,\circ\, -x \,=\, ln(cosh(x)) \,+\, ln(2) \)