User blog:B1mb0w/Alpha Function Code Version 7

Alpha Function Code Version 7
WORK IN PROGRESS

Test Bed for Version 7
Below is the test bed and various results using version 7.

\(\alpha(0) = 0\)

\(\alpha(0.5) = 1\)

\(\alpha(1) = 2\)

\(\alpha(1.5) = f_{0}(2)\)

\(\alpha(1.75) = (f_{1}(2))\)

\(\alpha(2) = f_{0}(f_{1}(2))\)

\(\alpha(2.25) = f_{0}^2(f_{1}(2))\)

\(\alpha(2.5) = f_{0}^{f_{0}(2)}(f_{1}(2))\)

\(\alpha(3) = (f_{1}^{f_{0}(f_{1}(2))})(f_{\omega}(2))\)

\(\alpha(3.25) = f_{0}^2(f_{f_{0}(\omega)}(2))\)

\(\alpha(3.5) = (f_{2}^{f_{0}(f_{1})(f_{f_{0}(\omega)}(2))})(f_{\omega})(f_{f_{0}(\omega)}(2))\)

\(\alpha(3.75) = f_{0}(f_{1})(f_{f_{0}(f_{1}(2))}^{f_{0}^{f_{0}(2)}(f_{\omega}(2))})(f_{(f_{1}(\omega))}(2))\)

\(\alpha(4) = f_{0}^{f_{0}(f_{1}(2))}(f_{1})(f_{2})(f_{f_{0}(2)})(f_{f_{0}(f_{1}(2))})(f_{f_{0}(f_{1}(\omega))}(2))\)

\(\alpha(4.25) = f_{0}^2(f_{f_{0}^{f_{0}(\omega)}(f_{1}(\omega))}(2))\)

\(\alpha(4.5) = (f_{1})(f_{2})(f_{f_{0}(2)})(f_{(f_{1}(2))}^{f_{0}^2(f_{\omega}(2))})(f_{f_{0}(f_{1}(2))})(f_{f_{0}^{f_{0}(f_{1})(f_{\omega}(\omega))}(f_{1}(\omega))}(2))\)

\(\alpha(4.75) = (f_{f_{0}^{f_{0}^{f_{0}^{(f_{1}(\omega))}(f_{1})(f_{\omega})(f_{f_{0}(\omega)}(\omega))}(f_{\omega}^{f_{0}^{f_{0}^{f_{0}(\omega)}(f_{1}(\omega))}(f_{1})(f_{f_{0}(\omega)})(f_{\omega}(\omega))})(f_{f_{0}(\omega)}(\omega))}(f_{1}(\omega))}(2))\)