I want to get real solutions $a$ and $b$ to the equation
(2.40008*10^19 - 4.49878*10^19 I) E^(-2 I a -
2 b) ((0. + 0. I) - (4.5635*10^-23 + 8.15149*10^-23 I) E^(
4 I a) - (1.12075*10^-23 + 1.24058*10^-22 I) E^(
4 b) - (1.74844*10^-22 - 6.15711*10^-22 I) E^(
3 I a + b) + (8.94669*10^-22 - 7.38525*10^-22 I) E^(
2 I a + 2 b) - (3.46096*10^-22 - 6.52983*10^-22 I) E^(
I a + 3 b)) == 0
I try to do it in this way
NSolve[(2.4000786897488364`*^19 -
4.498778537609138`*^19 I) E^(-2 I a -
2 b) ((0.` +
0.` I) - (4.563496732920858`*^-23 +
8.151489821720291`*^-23 I) E^(
4 I a) - (1.120754493395002`*^-23 +
1.240575949143422`*^-22 I) E^(
4 b) - (1.7484395154776395`*^-22 -
6.157113597162467`*^-22 I) E^(
3 I a + b) + (8.9466944569964`*^-22 -
7.385245125236398`*^-22 I) E^(
2 I a + 2 b) - (3.4609629128991235`*^-22 -
6.529826731694082`*^-22 I) E^(I a + 3 b)) == 0, {a, b}, Reals]
but it returns a warning that all coefficients in the equation must be real. Then I try
NSolve[(2.4000786897488364`*^19 -
4.498778537609138`*^19 I) E^(-2 I a -
2 b) ((0.` +
0.` I) - (4.563496732920858`*^-23 +
8.151489821720291`*^-23 I) E^(
4 I a) - (1.120754493395002`*^-23 +
1.240575949143422`*^-22 I) E^(
4 b) - (1.7484395154776395`*^-22 -
6.157113597162467`*^-22 I) E^(
3 I a + b) + (8.9466944569964`*^-22 -
7.385245125236398`*^-22 I) E^(
2 I a + 2 b) - (3.4609629128991235`*^-22 -
6.529826731694082`*^-22 I) E^(I a + 3 b)) == 0 && Im[a] == 0 && Im[b] == 0, {a, b}]
The output is It seems that it is still adding complex numbers to the solution. What can I do now to get purely real solutions?
If we plot the two regions where the real and imaginary parts of the LHS are greater than $0$. We have the following graph, which shows that we have discrete solutions (where the boundaries intersect), so there is a solution near $a=0.9, b=-1.2$, as well as its antipode.