Dear Marco,
Thank you very much for your advice. I have followed them and I still have few to ask.
If I get k[y]
from this,
p = 1.57516; a1 = 1.7824; \[Alpha] = 1/2 Pi;
f[k_?NumericQ] :=
p*NIntegrate[
1/Sqrt[(1 - t) (1 - t p^2 Cos[\[Alpha]]) - 3 p t Log[t]], {t, k/p,
1}]
ky = FindRoot[y == f[k], {k, 0}];
Table[{y, ky[[1, 2]]}, {y, 0, a1 + \[Delta], \[Delta]}] // N
and I will get this
{{0., 1.57516 + 0. I}, {0.05, 1.57289 + 2.44473*10^-16 I}, {0.1,
1.56608 - 5.15215*10^-17 I}, {0.15,
1.55475 + 3.28431*10^-14 I}, {0.2,
1.53893 - 1.5182*10^-13 I}, {0.25,
1.51865 - 2.71443*10^-15 I}, {0.3,
1.49396 + 2.74854*10^-16 I}, {0.35, 1.46494}, {0.4, 1.43164}, {0.45,
1.39415}, {0.5, 1.35259}, {0.55, 1.30705}, {0.6, 1.25767}, {0.65,
1.20459}, {0.7, 1.14797}, {0.75, 1.088}, {0.8, 1.02488}, {0.85,
0.958821}, {0.9, 0.890087}, {0.95, 0.818952}, {1., 0.74573}, {1.05,
0.670772}, {1.1, 0.594476}, {1.15, 0.517297}, {1.2,
0.439757}, {1.25, 0.362468}, {1.3, 0.286157}, {1.35,
0.211719}, {1.4, 0.140297}, {1.45, 0.073469}, {1.5,
0.0138314}, {1.55, -0.0326749 +
1.68698*10^-15 I}, {1.6, -0.0692389 -
3.06896*10^-15 I}, {1.65, -0.136826 -
1.66952*10^-15 I}, {1.7, -0.186826 -
2.87141*10^-15 I}, {1.75, -0.236826 -
2.16633*10^-15 I}, {1.8, -0.286826 - 3.02064*10^-15 I}}
I suppose that the ky[[1, 2]]
as k[y]
will be used in the stream function
\[Psi][y_, z_] := (
Sqrt[(1 - k[y]/p) (1 - k[y] p Cos[\[Alpha]] - k[y] Log[k[y]/p])]
z^2 (k[y] - z))/(8 k[y] );
but I have problem to apply it. Do you have any suggestion how can I extract the ky[[1, 2]]
to be k[y]
? Thanks in advance :D