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.

complex fitting

Anton Antonov · Answer

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}]

JimB · Answer

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.

Complex Data Fitting

2 Answers

Separate fitting of real and imaginary parts

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