According to the code / calculations below, it seems that you only need higher values to `MaxRecursion`

. Then only slow convergence messages are given ("NIntegrate :: slwcon").

```
a = 7; m = 5; n = 1;
Print("nEquation: z^", m, " - ", a, "*z^", n, " - 1 = 0n");
Print("Ordinary solution:");
NSolve((z^m - a z^n - 1))
sol = z /. NSolve((z^m - a z^n - 1))
(* During evaluation of In(88):=
Equation: z^5 - 7*z^1 - 1 = 0
During evaluation of In(88):= Ordinary solution: *)
(* {{z -> -1.58871}, {z -> -0.142866}, {z ->
0.0355442 - 1.62852 I}, {z -> 0.0355442 + 1.62852 I}, {z ->
1.66049}}
{-1.58871, -0.142866, 0.0355442 - 1.62852 I,
0.0355442 + 1.62852 I, 1.66049} *)
Print("Solution with definite integration:"); S =
Table(Exp(2 j Pi I/m) +
1/(2 Pi I) (Exp((2 j + 1) Pi I/m)*
NIntegrate(
Log(1 + a t^n/(1 + t^m) Exp((2 j + 1) Pi I n/m)), {t, 0,
Infinity}, MaxRecursion -> 200) -
Exp((2 j - 1) Pi I/m)*
NIntegrate(
Log(1 + a t^n/(1 + t^m) Exp((2 j - 1) Pi I n/m)), {t, 0,
Infinity}, MaxRecursion -> 200)), {j, 0, m - 1});
(*
During evaluation of In(93):= Solution with definite integration:
During evaluation of In(93):= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
During evaluation of In(93):= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. *)
S
(* {1.66049 + 0. I,
0.0355442 + 1.62852 I, -1.58871 + 7.8*10^-8 I, -0.142866 -
7.8*10^-8 I, 0.0355442 - 1.62852 I} *)
```

Here we see that you have all the values in your "ordinary solution" (using a certain tolerance):

```
Complement(S, sol,
SameTest -> (Abs(#1 - #2)/Norm({#1, #2}, Infinity) < 10^-6 &))
(* {} *)
```