Complex Data Fitting

Question

I have a complex data set and I want to fit with a mathematical model using NonlinearModelFit, but an error appears. Following is the data I'm using and the code that I have tried.

 dataset = {{0.1, 78.2356 - 0.8683 I}, {0.1998, 78.4897 - 0.5794 I}, {0.2996, 
      78.1675 - 1.0737 I}, {0.3994, 78.2344 - 1.4989 I}, {0.4992, 
      78.1883 - 1.8812 I}, {0.599, 78.1934 - 2.2289 I}, {0.6988, 
      78.1711 - 2.5836 I}, {0.7986, 78.1692 - 3.0154 I}, {0.8984, 
      78.1315 - 3.3294 I}, {0.9982, 78.0316 - 3.6955 I}, {1.098, 
      78.0287 - 4.0985 I}, {1.1978, 77.9932 - 4.4796 I}, {1.2976, 
      77.9667 - 4.8723 I}, {1.3974, 77.8871 - 5.2557 I}, {1.4972, 
      77.7778 - 5.6587 I}, {1.597, 77.5958 - 5.9334 I}, {1.6968, 
      77.7181 - 6.2427 I}, {1.7966, 77.661 - 6.6707 I}, {1.8964, 
      77.6011 - 7.0251 I}, {1.9962, 77.4963 - 7.394 I}, {2.096, 
      77.4624 - 7.8034 I}, {2.1958, 77.3749 - 8.1873 I}, {2.2956, 
      77.2339 - 8.5655 I}, {2.3954, 77.1747 - 8.8844 I}, {2.4952, 
      76.9861 - 9.2161 I}, {2.595, 76.9674 - 9.4487 I}, {2.6948, 
      77.0184 - 9.9153 I}, {2.7946, 76.8488 - 10.3242 I}, {2.8944, 
      76.7229 - 10.6691 I}, {2.9942, 76.6241 - 11.0533 I}, {3.094, 
      76.4483 - 11.3778 I}, {3.1938, 76.3633 - 11.7194 I}, {3.2936, 
      76.2347 - 12.0944 I}, {3.3934, 76.117 - 12.3783 I}, {3.4932, 
      76.0038 - 12.7369 I}, {3.593, 75.897 - 13.1024 I}, {3.6928, 
      75.7642 - 13.4812 I}, {3.7926, 75.6128 - 13.8315 I}, {3.8924, 
      75.4548 - 14.1748 I}, {3.9922, 75.3121 - 14.5079 I}, {4.092, 
      75.112 - 14.846 I}, {4.1918, 74.9628 - 15.1255 I}, {4.2916, 
      74.7993 - 15.4578 I}, {4.3914, 74.6315 - 15.8132 I}, {4.4912, 
      74.4547 - 16.1097 I}, {4.591, 74.2938 - 16.4089 I}, {4.6908, 
      74.152 - 16.7319 I}, {4.7906, 73.9702 - 16.9752 I}, {4.8904, 
      73.8144 - 17.2465 I}, {4.9902, 73.6398 - 17.5256 I}, {5.09, 
      73.5109 - 17.8629 I}, {5.1898, 73.3688 - 18.1599 I}, {5.2896, 
      73.2115 - 18.4785 I}, {5.3894, 73.0502 - 18.7804 I}, {5.4892, 
      72.8459 - 19.0808 I}, {5.589, 72.6908 - 19.4046 I}, {5.6888, 
      72.5409 - 19.711 I}, {5.7886, 72.3442 - 20.0245 I}, {5.8884, 
      72.1652 - 20.2782 I}, {5.9882, 71.9966 - 20.6356 I}, {6.088, 
      71.7659 - 20.9083 I}, {6.1878, 71.5486 - 21.2 I}, {6.2876, 
      71.3843 - 21.4641 I}, {6.3874, 71.1986 - 21.7711 I}, {6.4872, 
      70.9554 - 22.0703 I}, {6.587, 70.724 - 22.3318 I}, {6.6868, 
      70.4961 - 22.6017 I}, {6.7866, 70.2746 - 22.8812 I}, {6.8864, 
      70.0688 - 23.1126 I}, {6.9862, 69.8475 - 23.4055 I}, {7.086, 
      69.6181 - 23.6201 I}, {7.1858, 69.4073 - 23.8674 I}, {7.2856, 
      69.1847 - 24.1005 I}, {7.3854, 68.9453 - 24.3616 I}, {7.4852, 
      68.7403 - 24.5436 I}, {7.585, 68.5239 - 24.8114 I}, {7.6848, 
      68.2914 - 25.0535 I}, {7.7846, 68.0972 - 25.2649 I}, {7.8844, 
      67.8553 - 25.5043 I}, {7.9842, 67.6882 - 25.7208 I}, {8.084, 
      67.4227 - 25.9499 I}, {8.1838, 67.1762 - 26.1625 I}, {8.2836, 
      66.9512 - 26.3742 I}, {8.3834, 66.7394 - 26.6041 I}, {8.4832, 
      66.5167 - 26.842 I}, {8.583, 66.3146 - 27.0334 I}, {8.6828, 
      66.0724 - 27.2423 I}, {8.7826, 65.8313 - 27.4527 I}, {8.8824, 
      65.6043 - 27.6363 I}, {8.9822, 65.3769 - 27.9247 I}, {9.082, 
      65.1264 - 28.1023 I}, {9.1818, 64.8719 - 28.2838 I}, {9.2816, 
      64.6268 - 28.4543 I}, {9.3814, 64.4335 - 28.6643 I}, {9.4812, 
      64.1462 - 28.892 I}, {9.581, 63.9215 - 29.021 I}, {9.6808, 
      63.6482 - 29.2224 I}, {9.7806, 63.4241 - 29.4026 I}, {9.8804, 
      63.2 - 29.5862 I}, {9.9802, 62.9566 - 29.749 I}, {10.08, 
      62.7357 - 29.9395 I}, {10.1798, 62.4715 - 30.1185 I}, {10.2796, 
      62.1795 - 30.2156 I}, {10.3794, 61.96 - 30.4348 I}, {10.4792, 
      61.7823 - 30.5786 I}, {10.579, 61.4458 - 30.7737 I}, {10.6788, 
      61.2525 - 30.9524 I}, {10.7786, 60.9821 - 31.029 I}, {10.8784, 
      60.7528 - 31.2795 I}, {10.9782, 60.497 - 31.3811 I}, {11.078, 
      60.2911 - 31.4382 I}, {11.1778, 60.0138 - 31.6786 I}, {11.2776, 
      59.7766 - 31.7127 I}, {11.3774, 59.5339 - 31.9257 I}, {11.4772, 
      59.3229 - 32.0314 I}, {11.577, 59.0414 - 32.1797 I}, {11.6768, 
      58.7069 - 32.3222 I}, {11.7766, 58.6335 - 32.4309 I}, {11.8764, 
      58.2855 - 32.6102 I}, {11.9762, 57.9934 - 32.5999 I}, {12.076, 
      57.8657 - 32.8212 I}, {12.1758, 57.5263 - 32.79 I}, {12.2756, 
      57.43 - 33.0486 I}, {12.3754, 56.9925 - 33.0071 I}, {12.4752, 
      56.9032 - 33.357 I}, {12.575, 56.6686 - 33.2642 I}, {12.6748, 
      56.4106 - 33.4076 I}, {12.7746, 56.1114 - 33.6666 I}, {12.8744, 
      55.9125 - 33.7541 I}, {12.9742, 55.7078 - 33.7603 I}, {13.074, 
      55.3646 - 33.919 I}, {13.1738, 55.1957 - 33.9628 I}, {13.2736, 
      54.8702 - 34.0163 I}, {13.3734, 54.6922 - 34.2249 I}, {13.4732, 
      54.4734 - 34.3232 I}, {13.573, 54.2027 - 34.3559 I}, {13.6728, 
      53.9333 - 34.4731 I}, {13.7726, 53.6691 - 34.5305 I}, {13.8724, 
      53.5039 - 34.4862 I}, {13.9722, 53.27 - 34.6338 I}, {14.072, 
      52.8795 - 34.7439 I}, {14.1718, 52.6861 - 34.7778 I}, {14.2716, 
      52.503 - 34.8645 I}, {14.3714, 52.4029 - 34.9377 I}, {14.4712, 
      52.2769 - 35.0561 I}, {14.571, 51.9974 - 35.1729 I}, {14.6708, 
      51.6274 - 35.2887 I}, {14.7706, 51.3093 - 35.3206 I}, {14.8704, 
      51.1775 - 35.2592 I}, {14.9702, 50.9441 - 35.2813 I}, {15.07, 
      50.6445 - 35.2355 I}, {15.1698, 50.387 - 35.4794 I}, {15.2696, 
      50.2707 - 35.462 I}, {15.3694, 49.9586 - 35.5747 I}, {15.4692, 
      49.6906 - 35.4405 I}, {15.569, 49.5962 - 35.6865 I}, {15.6688, 
      49.2882 - 35.7023 I}, {15.7686, 48.9925 - 35.7119 I}, {15.8684, 
      48.9255 - 35.7546 I}, {15.9682, 48.8472 - 35.9351 I}, {16.068, 
      48.4388 - 35.908 I}, {16.1678, 48.4093 - 35.9757 I}, {16.2676, 
      48.0541 - 35.8649 I}, {16.3674, 47.932 - 35.9882 I}, {16.4672, 
      47.6193 - 35.9926 I}, {16.567, 47.423 - 36.2149 I}, {16.6668, 
      47.1304 - 36.1237 I}, {16.7666, 46.8657 - 36.0961 I}, {16.8664, 
      46.8233 - 36.1513 I}, {16.9662, 46.4927 - 36.3335 I}, {17.066, 
      46.282 - 36.1248 I}, {17.1658, 46.0527 - 36.2736 I}, {17.2656, 
      45.7096 - 36.1602 I}, {17.3654, 45.6516 - 36.1769 I}, {17.4652, 
      45.5632 - 36.3304 I}, {17.565, 45.2815 - 36.3742 I}, {17.6648, 
      45.1287 - 36.4635 I}, {17.7646, 44.8639 - 36.384 I}, {17.8644, 
      44.5912 - 36.4433 I}, {17.9642, 44.3884 - 36.4687 I}, {18.064, 
      44.2389 - 36.4418 I}, {18.1638, 44.1411 - 36.5143 I}, {18.2636, 
      43.9103 - 36.559 I}, {18.3634, 43.6563 - 36.4571 I}, {18.4632, 
      43.4871 - 36.5651 I}, {18.563, 43.3273 - 36.5756 I}, {18.6628, 
      42.9873 - 36.6038 I}, {18.7626, 42.9083 - 36.5295 I}, {18.8624, 
      42.6972 - 36.3761 I}, {18.9622, 42.3707 - 36.5342 I}, {19.062, 
      42.2738 - 36.5439 I}, {19.1618, 42.0595 - 36.5423 I}, {19.2616, 
      41.9375 - 36.4932 I}, {19.3614, 41.7015 - 36.5501 I}, {19.4612, 
      41.5701 - 36.4344 I}, {19.561, 41.2793 - 36.4456 I}, {19.6608, 
      41.0975 - 36.544 I}, {19.7606, 40.9272 - 36.5619 I}, {19.8604, 
      40.8173 - 36.4263 I}, {19.9602, 40.5276 - 36.4277 I}, {20.06, 
      40.5298 - 36.3943 I}, {20.1598, 40.2316 - 36.3789 I}, {20.2596, 
      40.1209 - 36.4193 I}, {20.3594, 39.8604 - 36.5787 I}, {20.4592, 
      39.7385 - 36.4455 I}, {20.559, 39.5266 - 36.522 I}, {20.6588, 
      39.3144 - 36.4394 I}, {20.7586, 39.3321 - 36.4143 I}, {20.8584, 
      39.0912 - 36.4885 I}, {20.9582, 38.9314 - 36.2492 I}, {21.058, 
      38.5343 - 36.24 I}, {21.1578, 38.5108 - 36.3822 I}, {21.2576, 
      38.4282 - 36.3555 I}, {21.3574, 38.1504 - 36.4566 I}, {21.4572, 
      38.0167 - 36.2047 I}, {21.557, 37.7961 - 36.2409 I}, {21.6568, 
      37.5403 - 36.2281 I}, {21.7566, 37.3251 - 36.2135 I}, {21.8564, 
      37.258 - 36.1749 I}, {21.9562, 37.174 - 36.1892 I}, {22.056, 
      37.0593 - 36.2818 I}, {22.1558, 36.7329 - 36.2821 I}, {22.2556, 
      36.6355 - 36.2482 I}, {22.3554, 36.466 - 36.1053 I}, {22.4552, 
      36.281 - 36.0756 I}, {22.555, 36.2407 - 36.1418 I}, {22.6548, 
      36.0274 - 36.1273 I}, {22.7546, 35.9257 - 36.0706 I}, {22.8544, 
      35.7085 - 36.0424 I}, {22.9542, 35.487 - 35.9645 I}, {23.054, 
      35.3758 - 35.8514 I}, {23.1538, 35.24 - 35.909 I}, {23.2536, 
      35.1003 - 36.0131 I}, {23.3534, 35.0614 - 35.9159 I}, {23.4532, 
      34.9312 - 35.866 I}, {23.553, 34.429 - 35.9661 I}, {23.6528, 
      34.4772 - 35.6764 I}, {23.7526, 34.1887 - 35.7948 I}, {23.8524, 
      34.0702 - 35.7791 I}, {23.9522, 34.0382 - 35.7418 I}, {24.052, 
      33.7731 - 35.8373 I}, {24.1518, 33.7543 - 35.5773 I}, {24.2516, 
      33.6515 - 35.5328 I}, {24.3514, 33.3638 - 35.7687 I}, {24.4512, 
      33.2197 - 35.5497 I}, {24.551, 33.1856 - 35.429 I}, {24.6508, 
      32.8109 - 35.4585 I}, {24.7506, 32.8435 - 35.3567 I}, {24.8504, 
      32.6808 - 35.4156 I}, {24.9502, 32.5051 - 35.3273 I}, {25.05, 
      32.2659 - 35.1744 I}, {25.1498, 32.4265 - 35.3344 I}, {25.2496, 
      32.1252 - 35.2248 I}, {25.3494, 31.9565 - 35.2524 I}, {25.4492, 
      32.0133 - 35.1773 I}, {25.549, 31.8092 - 35.1778 I}, {25.6488, 
      32.0878 - 35.0699 I}, {25.7486, 31.5873 - 34.6076 I}, {25.8484, 
      31.2771 - 35.0794 I}, {25.9482, 31.5152 - 34.5532 I}, {26.048, 
      31.303 - 34.9144 I}, {26.1478, 30.9818 - 34.9513 I}, {26.2476, 
      30.8336 - 34.773 I}, {26.3474, 31.0272 - 34.896 I}, {26.4472, 
      30.6672 - 34.7709 I}, {26.547, 30.5985 - 34.8338 I}, {26.6468, 
      30.5358 - 34.8296 I}, {26.7466, 30.2887 - 34.7839 I}, {26.8464, 
      30.2015 - 34.5392 I}, {26.9462, 30.0512 - 34.6196 I}, {27.046, 
      29.9653 - 34.541 I}, {27.1458, 29.804 - 34.5801 I}, {27.2456, 
      29.6835 - 34.5962 I}, {27.3454, 29.6406 - 34.502 I}, {27.4452, 
      29.4564 - 34.4098 I}, {27.545, 29.3452 - 34.4996 I}, {27.6448, 
      29.1932 - 34.3772 I}, {27.7446, 29.1953 - 34.2989 I}, {27.8444, 
      29.0024 - 34.1343 I}, {27.9442, 28.8577 - 34.1517 I}, {28.044, 
      28.703 - 34.0196 I}, {28.1438, 28.6418 - 34.0946 I}, {28.2436, 
      28.58 - 34.0431 I}, {28.3434, 28.3131 - 33.9986 I}, {28.4432, 
      28.2246 - 33.9749 I}, {28.543, 28.1047 - 34.0503 I}, {28.6428, 
      28.0789 - 33.7741 I}, {28.7426, 27.8409 - 33.7681 I}, {28.8424, 
      27.8007 - 33.8519 I}, {28.9422, 27.5743 - 33.7117 I}, {29.042, 
      27.8188 - 33.6009 I}, {29.1418, 27.6161 - 33.5073 I}, {29.2416, 
      27.4297 - 33.6119 I}, {29.3414, 27.3634 - 33.5724 I}, {29.4412, 
      27.3038 - 33.5951 I}, {29.541, 27.1626 - 33.2034 I}, {29.6408, 
      27.0371 - 33.3796 I}, {29.7406, 26.8387 - 33.268 I}, {29.8404, 
      26.8122 - 33.2622 I}, {29.9402, 26.6344 - 33.2716 I}, {30.04, 
      26.6695 - 33.3534 I}, {30.1398, 26.5725 - 32.789 I}, {30.2396, 
      26.3733 - 33.226 I}, {30.3394, 26.2388 - 32.951 I}, {30.4392, 
      26.5557 - 33.177 I}, {30.539, 26.1338 - 33.0525 I}, {30.6388, 
      25.8534 - 32.933 I}, {30.7386, 26.0376 - 32.9274 I}, {30.8384, 
      26.0954 - 32.9956 I}, {30.9382, 25.7592 - 32.9014 I}, {31.038, 
      25.4958 - 32.8915 I}, {31.1378, 25.6263 - 32.6875 I}, {31.2376, 
      25.3777 - 32.7197 I}, {31.3374, 25.4505 - 32.852 I}, {31.4372, 
      25.2404 - 32.7977 I}, {31.537, 25.3987 - 32.4925 I}, {31.6368, 
      25.1363 - 32.3624 I}, {31.7366, 24.9608 - 32.3569 I}, {31.8364, 
      24.6876 - 32.5949 I}, {31.9362, 24.8773 - 32.361 I}, {32.036, 
      24.7508 - 32.3924 I}, {32.1358, 24.7399 - 32.16 I}, {32.2356, 
      24.5998 - 32.0557 I}, {32.3354, 24.5433 - 32.1212 I}, {32.4352, 
      24.3455 - 32.3712 I}, {32.535, 23.9694 - 32.0721 I}, {32.6348, 
      24.1629 - 31.8856 I}, {32.7346, 24.0315 - 32.056 I}, {32.8344, 
      23.7825 - 31.8144 I}, {32.9342, 24.0951 - 31.867 I}, {33.034, 
      23.6539 - 31.6875 I}, {33.1338, 23.7025 - 31.7136 I}, {33.2336, 
      23.5243 - 31.7632 I}, {33.3334, 23.6693 - 31.6976 I}, {33.4332, 
      23.2781 - 32.0396 I}, {33.533, 23.2774 - 31.6915 I}, {33.6328, 
      23.4158 - 31.588 I}, {33.7326, 23.2458 - 31.0499 I}, {33.8324, 
      23.385 - 31.3526 I}, {33.9322, 23.1822 - 31.9303 I}, {34.032, 
      23.0974 - 30.9541 I}, {34.1318, 22.9448 - 30.8666 I}, {34.2316, 
      22.9013 - 31.1484 I}, {34.3314, 23.2542 - 31.2115 I}, {34.4312, 
      22.397 - 31.0947 I}, {34.531, 22.9416 - 31.5076 I}, {34.6308, 
      21.9635 - 31.074 I}, {34.7306, 22.3843 - 31.0943 I}, {34.8304, 
      22.3009 - 31.1258 I}, {34.9302, 22.1042 - 31.1654 I}, {35.03, 
      22.5128 - 30.9664 I}, {35.1298, 22.339 - 31.1127 I}, {35.2296, 
      21.9851 - 30.8603 I}, {35.3294, 22.1375 - 30.9703 I}, {35.4292, 
      21.9631 - 30.7983 I}, {35.529, 22.286 - 30.9338 I}, {35.6288, 
      21.458 - 30.3835 I}, {35.7286, 21.3285 - 30.7709 I}, {35.8284, 
      21.8479 - 30.4151 I}, {35.9282, 21.6401 - 30.5023 I}, {36.028, 
      21.6042 - 30.496 I}, {36.1278, 21.461 - 30.0798 I}, {36.2276, 
      21.2528 - 29.9247 I}, {36.3274, 21.2585 - 30.5237 I}, {36.4272, 
      21.3279 - 30.3249 I}, {36.527, 21.4263 - 30.1211 I}, {36.6268, 
      21.4058 - 30.3662 I}, {36.7266, 21.1253 - 30.1514 I}, {36.8264, 
      21.2413 - 30.4202 I}, {36.9262, 21.1598 - 30.0798 I}, {37.026, 
      20.9465 - 30.0659 I}, {37.1258, 20.6829 - 30.0269 I}, {37.2256, 
      20.6885 - 29.8712 I}, {37.3254, 20.4891 - 30.1871 I}, {37.4252, 
      20.7532 - 30.1849 I}, {37.525, 20.7161 - 29.8535 I}, {37.6248, 
      20.735 - 29.5632 I}, {37.7246, 20.2155 - 29.6935 I}, {37.8244, 
      20.0769 - 29.622 I}, {37.9242, 20.4147 - 29.9926 I}, {38.024, 
      20.9966 - 29.8357 I}, {38.1238, 20.4021 - 29.9879 I}, {38.2236, 
      20.136 - 29.2182 I}, {38.3234, 20.0021 - 29.7264 I}, {38.4232, 
      20.063 - 29.5102 I}, {38.523, 19.7925 - 29.852 I}, {38.6228, 
      20.2011 - 29.744 I}, {38.7226, 19.6156 - 29.6467 I}, {38.8224, 
      19.9306 - 28.4294 I}, {38.9222, 19.3085 - 28.5398 I}, {39.022, 
      19.1026 - 29.2675 I}, {39.1218, 19.1628 - 29.3752 I}, {39.2216, 
      19.605 - 28.9693 I}, {39.3214, 19.6677 - 28.804 I}, {39.4212, 
      18.8673 - 28.5751 I}, {39.521, 18.5662 - 28.675 I}, {39.6208, 
      18.6279 - 28.8541 I}, {39.7206, 18.8227 - 28.9262 I}, {39.8204, 
      19.4427 - 28.9469 I}, {39.9202, 19.5005 - 28.3647 I}, {40.02, 
      18.4979 - 28.1689 I}, {40.1198, 18.2211 - 28.3022 I}, {40.2196, 
      18.569 - 28.2742 I}, {40.3194, 18.7859 - 28.6754 I}, {40.4192, 
      18.5712 - 28.4534 I}, {40.519, 19.3936 - 27.9846 I}, {40.6188, 
      18.4431 - 27.5675 I}, {40.7186, 18.1889 - 28.0742 I}, {40.8184, 
      18.5421 - 28.2055 I}, {40.9182, 18.6898 - 28.4374 I}, {41.018, 
      18.3297 - 28.0645 I}, {41.1178, 18.8042 - 27.8506 I}, {41.2176, 
      18.3059 - 27.344 I}, {41.3174, 18.5508 - 27.7339 I}, {41.4172, 
      17.9299 - 28.1919 I}, {41.517, 17.9836 - 27.7716 I}, {41.6168, 
      18.2556 - 27.7537 I}, {41.7166, 17.9775 - 28.0211 I}, {41.8164, 
      18.3221 - 27.786 I}, {41.9162, 17.7802 - 27.4491 I}, {42.016, 
      17.9248 - 27.6812 I}, {42.1158, 17.9283 - 27.5206 I}, {42.2156, 
      18.1385 - 27.7968 I}, {42.3154, 17.5093 - 27.5915 I}, {42.4152, 
      18.0288 - 27.3542 I}, {42.515, 17.8391 - 27.5258 I}, {42.6148, 
      18.2251 - 27.4179 I}, {42.7146, 17.4823 - 27.427 I}, {42.8144, 
      17.5969 - 27.0717 I}, {42.9142, 17.2438 - 27.2268 I}, {43.014, 
      17.5722 - 26.9162 I}, {43.1138, 17.0512 - 27.4752 I}, {43.2136, 
      17.731 - 27.7474 I}, {43.3134, 17.9637 - 27.0977 I}, {43.4132, 
      17.8519 - 26.5461 I}, {43.513, 16.4758 - 27.4136 I}, {43.6128, 
      17.5664 - 27.4024 I}, {43.7126, 17.3127 - 26.7502 I}, {43.8124, 
      17.0702 - 28.1142 I}, {43.9122, 16.944 - 25.6304 I}, {44.012, 
      17.6921 - 26.3435 I}, {44.1118, 16.7857 - 26.9177 I}, {44.2116, 
      17.8315 - 26.765 I}, {44.3114, 16.7895 - 26.8332 I}, {44.4112, 
      16.9196 - 26.2703 I}, {44.511, 16.7878 - 26.4833 I}, {44.6108, 
      16.5801 - 26.8623 I}, {44.7106, 16.9558 - 26.4815 I}, {44.8104, 
      16.7723 - 26.5304 I}, {44.9102, 16.6461 - 26.5571 I}, {45.01, 
      16.445 - 26.2377 I}, {45.1098, 16.3595 - 26.1725 I}, {45.2096, 
      16.6057 - 26.3542 I}, {45.3094, 16.8819 - 26.7169 I}, {45.4092, 
      16.7706 - 26.4544 I}, {45.509, 16.186 - 26.0565 I}, {45.6088, 
      16.5001 - 26.0794 I}, {45.7086, 16.2941 - 26.4279 I}, {45.8084, 
      16.0685 - 26.4002 I}, {45.9082, 16.5396 - 26.0268 I}, {46.008, 
      15.7409 - 26.6186 I}, {46.1078, 16.4295 - 26.1331 I}, {46.2076, 
      15.5538 - 26.457 I}, {46.3074, 16.4832 - 25.7581 I}, {46.4072, 
      15.8374 - 26.0055 I}, {46.507, 15.7819 - 26.8736 I}, {46.6068, 
      15.8253 - 25.7855 I}, {46.7066, 16.385 - 26.3928 I}, {46.8064, 
      16.2869 - 26.1208 I}, {46.9062, 15.5076 - 25.7764 I}, {47.006, 
      15.9992 - 26.1976 I}, {47.1058, 15.8232 - 25.8068 I}, {47.2056, 
      15.6416 - 25.5574 I}, {47.3054, 15.7279 - 25.4791 I}, {47.4052, 
      15.23 - 24.9536 I}, {47.505, 16.2428 - 24.9651 I}, {47.6048, 
      15.4889 - 26.3522 I}, {47.7046, 15.2959 - 24.9988 I}, {47.8044, 
      15.3342 - 25.6946 I}, {47.9042, 15.3437 - 25.544 I}, {48.004, 
      15.3685 - 25.4386 I}, {48.1038, 15.2888 - 25.6183 I}, {48.2036, 
      15.6667 - 25.3608 I}, {48.3034, 15.5084 - 24.8011 I}, {48.4032, 
      15.3141 - 25.2034 I}, {48.503, 15.0334 - 25.3849 I}, {48.6028, 
      15.3935 - 24.5825 I}, {48.7026, 15.4125 - 25.0509 I}, {48.8024, 
      14.9718 - 25.1981 I}, {48.9022, 14.7825 - 25.0575 I}, {49.002, 
      15.2234 - 25.1159 I}, {49.1018, 14.9325 - 25.1141 I}, {49.2016, 
      14.9938 - 25.3016 I}, {49.3014, 14.9617 - 24.3614 I}, {49.4012, 
      14.8526 - 25.004 I}, {49.501, 14.8749 - 24.9538 I}, {49.6008, 
      15.208 - 24.7995 I}, {49.7006, 14.6468 - 25.1923 I}, {49.8004, 
      14.7013 - 24.5138 I}, {49.9002, 15.0272 - 24.3538 I}, {50., 
      14.9279 - 24.6538 I}};

Here is the the code:

Subscript[ϵ, 0] = 8.854*10^-12
modelfit = 
  NonlinearModelFit[dataset, 
    Subscript[ϵ, ∞] + (Subscript[ϵ, 1] - 
    Subscript[ϵ, ∞])/(
    1 + I*2*π*f*Subscript[τ, 1]) + σ/(
    I*2*π*f*Subscript[ϵ, 0]), {{Subscript[ϵ, 1],
    78.88}, {Subscript[ϵ, ∞], 
    5.2}, {Subscript[τ, 1], 0.008}, {σ, 0.2}}, f];

This error appears:

NonlinearModelFit::nrlnum: The function value {0.642538 -3.5951*10^10 I,0.382869
-1.79935*10^10 I,0.695794 -1.19997*10^10 I,<<45>>,0.867068 -7.35134*10^8 
I,0.878805 -7.20431*10^8 I,<<451>>} is not a list of real numbers with dimensions 
{501} at {Subscript[ϵ, 1],Subscript[ϵ, ∞],Subscript[τ, 1],σ} = {78.88,5.2,0.008,0.2}.

Can anyone please help me to get rid of this? Thank you.

Answer 1

Can you use model fitting that is done separately for the real and imaginary parts?

Example is given below.

Separate fitting of real and imaginary parts

Fit for the real parts:

e0 = 8.854*10^-12;
modelfitRe = NonlinearModelFit[Re /@ dataset, Re[eInf + (e1 - eInf)/(1 + I*2*\[Pi]*f*t1) + \[Sigma]/(I*2*\[Pi]*f*e0)], {{e1, 78.88}, {eInf, 5.2}, {t1, 0.008}, {\[Sigma], 0.2}},f]

Fit for the imaginary parts:

modelfitIm = NonlinearModelFit[Map[{#[[1]], Im[#[[2]]]} &, dataset], Im[eInf + (e1 - eInf)/(1 + I*2*\[Pi]*f*t1) + \[Sigma]/(I*2*\[Pi]*f*e0)], {{e1, 78.88}, {eInf, 5.2}, {t1, 0.008}, {\[Sigma], 0.2}},f]

NonlinearModelFit::sszero: The step size in the search has become less than the tolerance prescribed by the PrecisionGoal option, but the gradient is larger than the tolerance specified by the AccuracyGoal option. There is a possibility that the method has stalled at a point that is not a local minimum.

Make the complex numbers list of fits:

datasetFit = Map[{#, modelfitRe[#] + I*modelfitIm[#]} &, dataset[[All, 1]]];

See the errors for the real parts:

{ResourceFunction["RecordsSummary"][Abs[dataset[[All, 2]] - datasetFit[[All, 2]]]], ResourceFunction["RecordsSummary"][Abs[dataset[[All, 2]] - datasetFit[[All, 2]]]/Abs[dataset[[All, 2]]]]}

See the errors for the imaginary parts:

{ResourceFunction["RecordsSummary"][Abs[Im[dataset[[All, 2]] - datasetFit[[All, 2]]]]], ResourceFunction["RecordsSummary"][Abs[Im[dataset[[All, 2]] - datasetFit[[All, 2]]]/Im[dataset[[All, 2]]]]]}

Plot the original data and the fits:

gr1 = MapThread[
    ListPlot[#, PlotRange -> All, PlotTheme -> "Detailed", PlotLegends -> {"Original"}, PlotStyle -> {{PointSize[0.015], GrayLevel[0.7]}}, PlotLabel -> #2, ImageSize -> Large] &, {{Map[{#[[1]], Re[#[[2]]]} &, dataset], Map[{#[[1]], Im[#[[2]]]} &, dataset]}, {"Real parts", "Imaginary parts"}}];
gr2 = ListLinePlot[#, PlotRange -> All, PlotTheme -> "Detailed", PlotLegends -> {"Fit"}, PlotStyle -> {Red}, PlotLabel -> "Imaginary parts", ImageSize -> Large] & /@ {Map[{#[[1]], Re[#[[2]]]} &, datasetFit], Map[{#[[1]], Im[#[[2]]]} &, datasetFit]};
MapThread[Show, {gr1, gr2}]

Answer 2

I suspect that you need to have the same values of the parameters for both the real and imaginary parts. If the variances associated with the real and imaginary regressions are the same and the random errors are independent, then consider using MultiNonlinearModelFit.

If the variances are not known to be equal or the random errors are not independent, then consider the following. (Note that the above link the method to follow also assume that the errors have a constant variance which does not depend on among other things the predictor variable. More of that after looking at the residuals.)

The prediction equation can be separated into the real and imaginary parts (using @AntonAntonov 's notation)

eInf + (ϵ1 - eInf)/(1 + I 2 π f t1) + σ/(I 2 π f e0)// ComplexExpand

(* Create response and predicted matrices *)
response = Transpose[{Re[dataset[[All, 2]]], Im[dataset[[All, 2]]]}];
predicted = Transpose[{eInf + (e1 - eInf)/(1 + 4 dataset[[All, 1]]^2  π^2  t1^2),
    -(σ0/(2 dataset[[All, 1]] π)) + (2 (-e1 + eInf) dataset[[All, 1]] π t1)/(1 + 4 dataset[[All, 1]]^2 π^2 t1^2)}];

(* Log of the likelihood *)
logL = LogLikelihood[BinormalDistribution[{0, 0}, {σ1, σ2}, ρ], response - predicted];

(* Find maximum likelihood estimates *)
mle = FindMaximum[{logL, σ1 > 0 && σ2 > 0 && e1 > 0 && eInf > 0 && -1 < ρ < 1},
  {{e1, 78}, {eInf, 5}, {t1, 0.008}, {σ0, 0.1}, {σ1, 0.2}, {σ2, 0.2}, {ρ, 0}}]
(* {213.616, {e1 -> 78.2692, eInf -> 5.2173, t1 -> 0.00822744, σ0 -> 0.107965, 
  σ1 -> 0.185984, σ2 -> 0.205559, ρ -> 0.0164636}} *)

(* Get approximate 95% confidence intervals for the parameter estimates *)
covMat = -Inverse[(D[logL, {{e1, eInf, t1, σ0, σ1, σ2, ρ}, 2}]) /. mle[[2]]];
se = Sqrt[Diagonal[covMat]];
estimates = {e1, eInf, t1, σ0, σ1, σ2, ρ} /. mle[[2]];
lower95CL = ({e1, eInf, t1, σ0, σ1, σ2, ρ} /. mle[[2]]) - 1.96 se;
upper95CL = ({e1, eInf, t1, σ0, σ1, σ2, ρ} /. mle[[2]]) + 1.96 se;
TableForm[Transpose[{estimates, lower95CL, upper95CL}],
 TableHeadings -> {{"e1", "eInf", "t1", "σ0", "σ1", "σ2", "ρ"},
   {"Estimate", "Lower 95% CL", "Upper 95% CL"}}]

Now look at the residuals.

predresid = Join[Transpose[predicted], Transpose[response - predicted]] /. mle[[2]] // Transpose;
ListPlot[Transpose[{dataset[[All, 1]], predresid[[All, 3]]}], PlotStyle -> PointSize[0.01],
  AxesLabel -> (Style[#, Bold, 12] &) /@ {"f", "Residual"}, 
 PlotLabel -> Style["Real", 18, Bold], PlotRange -> All]
ListPlot[Transpose[{dataset[[All, 1]], predresid[[All, 4]]}], 
 PlotStyle -> PointSize[0.01],
 AxesLabel -> (Style[#, Bold, 12] &) /@ {"f", "Residual"}, 
 PlotLabel -> Style["Imaginary", 18, Bold], PlotRange -> All]

We see that the residuals are not constant with respect to the predictor variable $f$. If the residuals are small compared to the desired precision, then there's nothing to do. If the residuals are large compared to the desired precision, then one should also attempt to model the structure of the error variances. The actual estimates of the parameters likely won't change much but the associated standard errors and confidence intervals might depend strongly on adequately modeling the error structure.