\(F=e^{ i3\pi /4 }D\sqrt { 2{ mc^{ 2 } } } .tanh\left( \cfrac { { D } }{ \sqrt { 2{ mc^{ 2 } } } }( x-x_o).e^{ i\pi /4 } \right) \)
If \(G=D.e^{ i\pi /4 }\) is real then
\(F=e^{i\pi/2}\sqrt { 2{ mc^{ 2 } } }\,G.tanh\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_z) \right) \)
\(F=i\sqrt { 2{ mc^{ 2 } } }\,G.tanh\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_z) \right) \)
If however we allow the augment to the function \(tanh(x)\) to be complex, and interpret \(e^{ i\pi /4 }\) as rotation about the origin anti-clockwise by \(\pi/4\) from \( e^{ i\pi /4 } \) , and then a further rotation by \(3\pi/4\) from \( e^{ i3\pi /4 } \), \(F\) is rotated by \(\pi\) about the origin in total.
\(F=-\sqrt { 2{ mc^{ 2 } } }\,G.tanh\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_z) \right) \)
\(F\) is then along the negative \(x\) direction, towards the center of the particle which is consistent with the expression
\(4\pi { \cfrac { \dot { x } ^3 }{ a_{ \psi } } m }=-q\)
that a negative charge is normally associated with particle.
But, this interpretation of \(e^{ i\pi /4 }\) in the augment of \(tanh(x)\) is odd because \(tanh(x)\) does not accept complex augments. In this way, the modulus of a complex augment serves as the real input to the function and the argument of the complex augment rotates the x axis of the graph of the function about the origin anti-clockwise.
If this is true,
\(F=-\sqrt { 2{ mc^{ 2 } } }\,G.tanh\left( \cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } } (x-x_z) \right) \)
without \(i\). Force density \(F\), points towards the center of the particle, along the negative \(x\) direction.
And we have a new class of real functions with complex augments.
Good night.
Note: This suggests \(e^{ia}\) has immunity, that
\(F=e^{ i3\pi /4 }D\sqrt { 2{ mc^{ 2 } } } .tanh\left( \cfrac { { D } }{ \sqrt { 2{ mc^{ 2 } } } }( x-x_o).e^{ i\pi /4 } \right) \)
\(F=e^{ i3\pi /4 }.e^{ i\pi /4 } D\sqrt { 2{ mc^{ 2 } } } .tanh\left( \cfrac { { D } }{ \sqrt { 2{ mc^{ 2 } } } }( x-x_o)\right) \)
\(F=e^{ i\pi } D\sqrt { 2{ mc^{ 2 } } } .tanh\left( \cfrac { { D } }{ \sqrt { 2{ mc^{ 2 } } } }( x-x_o)\right) \)
\(F=e^{ i\pi } D\sqrt { 2{ mc^{ 2 } } } .tanh\left( \cfrac { { D } }{ \sqrt { 2{ mc^{ 2 } } } }( x-x_o)\right) \)
\(F=- D\sqrt { 2{ mc^{ 2 } } } .tanh\left( \cfrac { { D } }{ \sqrt { 2{ mc^{ 2 } } } }( x-x_o)\right) \)
\(e^{ia}\) passes through a function as a constant coefficient passes through an integral.
Warning, over generalization!!
