2
$\begingroup$

Let's define,

$k(e)=\sqrt{e}$, $\gamma_1(e,v_1,v_2)=(v_1+iv_2)/(i*k(e))$ , $\gamma_2(e,v_1,v_2)=(v_1-iv_2)/(i*k(e))$, $$\\$$ Now the function gets defined as, $f(e,v_1,v_2)=(2-\gamma_1(e,v_1,v_2)) (2-\gamma_2(e,v_1,v_2))-\gamma_1(e,v_1,v_2) \gamma_2(e,v_1,v_2)* e^{4 ik(e)}$

If I take, $v_1=1.4$, $v_2=2.2$, Then function $|f(e,v_1,v_2)|$ has a real root around 1.5, and I want know about its complex roots by any means.

Could you please help me ??

Here again rewrite above expressions,

k[e_]:=Sqrt[e];
gamma1[e_,v1_,v2_]:=(v1+I*v2)/(I*k[e]);
gamma2[e_,v1_,v2_]:=(v1-I*v2)/(I*k[e]);
f[e_,v1_,v2_]:=(2-gamma1[e,v1,v2])*(2-gamma2[e,v1,v2])-gamma1[e,v1,v2] *gamma2[e,v1,v2]* Exp[4*I*k[e]]
$\endgroup$
8
  • 3
    $\begingroup$ Provide your expressions as text-only Mathematica code in addition to $\LaTeX$ expressions. $\endgroup$
    – MarcoB
    May 6, 2018 at 18:29
  • $\begingroup$ @MarcoB thanks! Done $\endgroup$ May 6, 2018 at 18:34
  • $\begingroup$ That is not sensible Mathematica code. Misused parentheses and underscore, to start. $\endgroup$
    – John Doty
    May 6, 2018 at 18:36
  • $\begingroup$ @JohnDoty, oh! yes, corrected $\endgroup$ May 6, 2018 at 18:45
  • $\begingroup$ Ok, so how have you tried to solve it? $\endgroup$
    – John Doty
    May 6, 2018 at 18:58

2 Answers 2

3
$\begingroup$

Plotting suggests a root around 6-I.

ContourPlot[Abs[f[x + I y, 1.4, 2.2]], {x, 0, 10}, {y, -2, 10}]

enter image description here

FindRoot isn't very good for Abs[something]==0: it wants to see the function cross zero. Use FindMinimum:

FindMinimum[Abs[f[x + I y, 1.4, 2.2]], {{x, 6}, {y, -1}}]

After some complaining, it yields {1.30986*10^-7, {x -> 5.98286, y -> -1.3557}}, which may be good enough. Other roots, better roots, avoiding complaints, etc. left as an exercise for you.

$\endgroup$
5
$\begingroup$

If you bound the domain, NSolve can usually find all roots of an analytic function:

NSolve[f[e, 1.4, 2.2] == 0 && -10 < Re[e] < 10 && -10 < Im[e] < 10, e]
(*  {{e -> 1.46847 - 0.00315635 I}, {e -> 5.98286 - 1.3557 I}}  *)

You can make the domain somewhat larger:

NSolve[f[e, 1.4, 2.2] == 0 && 0 < Re[e] < 1000 && -100 < Im[e] < 100, e]
(*
{{e -> 1.46847 - 0.00315635 I}, {e -> 5.98286 - 1.3557 I}, {e -> 
   15.3143 - 4.22245 I}, {e -> 29.7131 - 7.79704 I}, {e -> 
   49.12 - 11.836 I}, {e -> 73.5089 - 16.229 I}, {e -> 
   102.866 - 20.9096 I}, {e -> 137.184 - 25.8326 I}, {e -> 
   176.457 - 30.9654 I}, {e -> 220.681 - 36.2831 I}, {e -> 
   269.854 - 41.7662 I}, {e -> 323.974 - 47.3988 I}, {e -> 
   383.038 - 53.1679 I}, {e -> 447.046 - 59.0626 I}, {e -> 
   515.996 - 65.0736 I}, {e -> 589.888 - 71.1928 I}, {e -> 
   668.721 - 77.4132 I}, {e -> 752.495 - 83.7288 I}, {e -> 
   841.208 - 90.134 I}, {e -> 934.86 - 96.6241 I}}
*)
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.