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Kramers-Kronig Relations, getting imaginary data from known real data but WITHOUT EQUATION

I am trying to get epsilon2 (imaginary part) from known epsilon1 (real part) data with the respective energy values (omega), using the Kramers-Kronig Relations.

This is the epsilon1 data:

epsilon1={{14.8294}, {12.8294}, {12.8299}, {12.8306}, {12.8314}, {12.8326}, \ {12.834}, {12.8356}, {12.8375}, {12.8396}, {12.842}, {12.8446}, \ {12.8475}, {12.8507}, {12.8541}, {12.8577}, {12.8617}, {12.8658}, \ {12.8702}, {12.8749}, {12.8799}, {12.8851}, {12.8905}, {12.8963}, \ {12.9022}, {12.9085}, {12.915}, {12.9218}, {12.9288}, {12.9361}, \ {12.9437}, {12.9516}, {12.9597}, {12.9681}, {12.9767}, {12.9857}, \ {12.9949}, {13.0044}, {13.0142}, {13.0243}, {13.0346}, {13.0453}, \ {13.0562}, {13.0674}, {13.0789}, {13.0907}, {13.1028}, {13.1152}, \ {13.1279}, {13.1409}, {13.1542}, {13.1679}, {13.1818}, {13.196}, \ {13.2106}, {13.2255}, {13.2407}, {13.2562}, {13.2721}, {13.2882}, \ {13.3048}, {13.3216}, {13.3388}, {13.3564}, {13.3743}, {13.3925}, \ {13.4111}, {13.4301}, {13.4494}, {13.4691}, {13.4891}, {13.5096}, \ {13.5304}, {13.5516}, {13.5731}, {13.5951}, {13.6175}, {13.6403}, \ {13.6635}, {13.6871}, {13.7111}, {13.7355}, {13.7604}, {13.7857}, \ {13.8114}, {13.8376}, {13.8643}, {13.8914}, {13.9189}, {13.947}, \ {13.9755}, {14.0045}, {14.034}, {14.064}, {14.0945}, {14.1255}, \ {14.157}, {14.1891}, {14.2217}, {14.2548}, {14.2885}, {14.3228}, \ {14.3576}, {14.3931}, {14.4291}, {14.4657}, {14.5029}, {14.5408}, \ {14.5793}, {14.6184}, {14.6582}, {14.6987}, {14.7398}, {14.7816}, \ {14.8242}, {14.8674}, {14.9114}, {14.9562}, {15.0016}, {15.0479}, \ {15.095}, {15.1428}, {15.1915}, {15.241}, {15.2914}, {15.3426}, \ {15.3947}, {15.4477}, {15.5017}, {15.5566}, {15.6124}, {15.6693}, \ {15.7271}, {15.786}, {15.8459}, {15.907}, {15.9691}, {16.0323}, \ {16.0967}, {16.1622}, {16.229}, {16.297}, {16.3662}, {16.4368}, \ {16.5086}, {16.5819}, {16.6565}, {16.7325}, {16.81}, {16.889}, \ {16.9695}, {17.0517}, {17.1354}, {17.2208}, {17.3079}, {17.3967}, \ {17.4874}, {17.5799}, {17.6743}, {17.7706}, {17.869}, {17.9694}, \ {18.072}, {18.1767}, {18.2837}, {18.3931}, {18.5048}, {18.619}, \ {18.7357}, {18.8551}, {18.9771}, {19.102}, {19.2297}, {19.3604}, \ {19.4942}, {19.6312}, {19.7714}, {19.915}, {20.0621}, {20.2129}, \ {20.3674}, {20.5258}, {20.6882}, {20.8547}, {21.0256}, {21.2009}, \ {21.3808}, {21.5655}, {21.7552}, {21.9501}, {22.1502}, {22.3559}, \ {22.5673}, {22.7847}, {23.0082}, {23.238}, {23.4745}, {23.7178}, \ {23.9682}, {24.2258}, {24.491}, {24.764}, {25.045}, {25.3342}, \ {25.6318}, {25.9381}, {26.253}, {26.5769}, {26.9096}, {27.2513}, \ {27.6018}, {27.9609}, {28.3282}, {28.7032}, {29.0852}, {29.473}, \ {29.8653}, {30.2601}, {30.6552}, {31.0474}, {31.433}, {31.807}, \ {32.1635}, {32.495}, {32.7928}, {33.0459}, {33.2419}, {33.3658}, \ {33.401}, {33.3293}, {33.1312}, {32.7876}, {32.2811}, {31.598}, \ {30.7315}, {29.6836}, {28.4678}, {27.11}, {25.6474}, {24.126}, \ {22.5953}, {21.1022}, {19.6847}, {18.3667}, {17.157}, {16.0507}, \ {15.0377}, {14.1135}, {13.2885}, {12.5862}, {12.0297}, {11.6249}, \ {11.3517}, {11.1678}, {11.0178}, {10.846}, {10.6145}, {10.3216}, \ {10.0032}, {9.70343}, {9.44313}, {9.21525}, {9.00088}, {8.7835}, \ {8.55443}, {8.31253}, {8.06202}, {7.81009}, {7.56502}, {7.33467}, \ {7.1254}, {6.94151}, {6.78485}, {6.65491}, {6.54901}, {6.46279}, \ {6.39076}, {6.32696}, {6.26554}, {6.20136}, {6.13033}, {6.04956}, \ {5.95728}, {5.85238}, {5.7337}, {5.59923}, {5.44513}, {5.26495}, \ {5.04901}, {4.78415}, {4.45383}, {4.03872}, {3.51771}, {2.86943}, \ {2.07425}, {1.11665}, {-0.0120996}, {-1.31138}, {-2.76918}, \ {-4.36103}, {-6.05058}, {-7.79182}, {-9.53297}, {-11.2214}, \ {-12.8084}, {-14.2537}, {-15.5279}, {-16.6136}, {-17.5047}, \ {-18.2047}, {-18.7244}, {-19.0794}, {-19.2882}, {-19.3702}, \ {-19.3443}, {-19.2286}, {-19.0393}, {-18.7909}, {-18.4958}, \ {-18.1648}, {-17.8071}, {-17.4301}, {-17.0403}, {-16.6429}, \ {-16.2423}, {-15.8421}, {-15.4452}, {-15.0542}, {-14.671}, \ {-14.2975}, {-13.9349}, {-13.5846}, {-13.2475}, {-12.9245}, \ {-12.6161}, {-12.3229}, {-12.0452}, {-11.783}, {-11.5362}, \ {-11.3044}, {-11.0871}, {-10.8835}, {-10.6926}, {-10.5132}, \ {-10.344}, {-10.1838}, {-10.0315}, {-9.88592}, {-9.7465}, {-9.61279}, \ {-9.48467}, {-9.36212}, {-9.24495}, {-9.13253}, {-9.02364}, \ {-8.9167}, {-8.81031}, {-8.70397}, {-8.59854}, {-8.49612}, \ {-8.39934}, {-8.31029}, {-8.22946}, {-8.15516}, {-8.08397}, \ {-8.01196}, {-7.93638}, {-7.85669}, {-7.77466}, {-7.69345}, \ {-7.61619}, {-7.5446}, {-7.47837}, {-7.41555}, {-7.35387}, \ {-7.29227}, {-7.23185}, {-7.17563}, {-7.12728}, {-7.08922}, \ {-7.06092}, {-7.03809}, {-7.01352}, {-6.97919}, {-6.9287}, \ {-6.85888}, {-6.76989}, {-6.66448}, {-6.54686}, {-6.42185}, \ {-6.29439}, {-6.16944}, {-6.05195}, {-5.94695}, {-5.85939}, \ {-5.79361}, {-5.75241}, {-5.73568}, {-5.73938}, {-5.75539}}

These are the respective energy values:

w={{0.}, {0.0139833}, {0.0279665}, {0.0419498}, {0.055933}, \ {0.0699163}, {0.0838995}, {0.0978828}, {0.111866}, {0.125849}, \ {0.139833}, {0.153816}, {0.167799}, {0.181782}, {0.195766}, \ {0.209749}, {0.223732}, {0.237715}, {0.251699}, {0.265682}, \ {0.279665}, {0.293648}, {0.307632}, {0.321615}, {0.335598}, \ {0.349581}, {0.363565}, {0.377548}, {0.391531}, {0.405514}, \ {0.419498}, {0.433481}, {0.447464}, {0.461447}, {0.475431}, \ {0.489414}, {0.503397}, {0.51738}, {0.531364}, {0.545347}, {0.55933}, \ {0.573313}, {0.587297}, {0.60128}, {0.615263}, {0.629247}, {0.64323}, \ {0.657213}, {0.671196}, {0.68518}, {0.699163}, {0.713146}, \ {0.727129}, {0.741113}, {0.755096}, {0.769079}, {0.783062}, \ {0.797046}, {0.811029}, {0.825012}, {0.838995}, {0.852979}, \ {0.866962}, {0.880945}, {0.894928}, {0.908912}, {0.922895}, \ {0.936878}, {0.950861}, {0.964845}, {0.978828}, {0.992811}, \ {1.00679}, {1.02078}, {1.03476}, {1.04874}, {1.06273}, {1.07671}, \ {1.09069}, {1.10468}, {1.11866}, {1.13264}, {1.14663}, {1.16061}, \ {1.17459}, {1.18858}, {1.20256}, {1.21654}, {1.23053}, {1.24451}, \ {1.25849}, {1.27248}, {1.28646}, {1.30044}, {1.31443}, {1.32841}, \ {1.34239}, {1.35638}, {1.37036}, {1.38434}, {1.39833}, {1.41231}, \ {1.42629}, {1.44028}, {1.45426}, {1.46824}, {1.48223}, {1.49621}, \ {1.51019}, {1.52418}, {1.53816}, {1.55214}, {1.56612}, {1.58011}, \ {1.59409}, {1.60807}, {1.62206}, {1.63604}, {1.65002}, {1.66401}, \ {1.67799}, {1.69197}, {1.70596}, {1.71994}, {1.73392}, {1.74791}, \ {1.76189}, {1.77587}, {1.78986}, {1.80384}, {1.81782}, {1.83181}, \ {1.84579}, {1.85977}, {1.87376}, {1.88774}, {1.90172}, {1.91571}, \ {1.92969}, {1.94367}, {1.95766}, {1.97164}, {1.98562}, {1.99961}, \ {2.01359}, {2.02757}, {2.04156}, {2.05554}, {2.06952}, {2.08351}, \ {2.09749}, {2.11147}, {2.12545}, {2.13944}, {2.15342}, {2.1674}, \ {2.18139}, {2.19537}, {2.20935}, {2.22334}, {2.23732}, {2.2513}, \ {2.26529}, {2.27927}, {2.29325}, {2.30724}, {2.32122}, {2.3352}, \ {2.34919}, {2.36317}, {2.37715}, {2.39114}, {2.40512}, {2.4191}, \ {2.43309}, {2.44707}, {2.46105}, {2.47504}, {2.48902}, {2.503}, \ {2.51699}, {2.53097}, {2.54495}, {2.55894}, {2.57292}, {2.5869}, \ {2.60089}, {2.61487}, {2.62885}, {2.64284}, {2.65682}, {2.6708}, \ {2.68479}, {2.69877}, {2.71275}, {2.72673}, {2.74072}, {2.7547}, \ {2.76868}, {2.78267}, {2.79665}, {2.81063}, {2.82462}, {2.8386}, \ {2.85258}, {2.86657}, {2.88055}, {2.89453}, {2.90852}, {2.9225}, \ {2.93648}, {2.95047}, {2.96445}, {2.97843}, {2.99242}, {3.0064}, \ {3.02038}, {3.03437}, {3.04835}, {3.06233}, {3.07632}, {3.0903}, \ {3.10428}, {3.11827}, {3.13225}, {3.14623}, {3.16022}, {3.1742}, \ {3.18818}, {3.20217}, {3.21615}, {3.23013}, {3.24412}, {3.2581}, \ {3.27208}, {3.28607}, {3.30005}, {3.31403}, {3.32802}, {3.342}, \ {3.35598}, {3.36996}, {3.38395}, {3.39793}, {3.41191}, {3.4259}, \ {3.43988}, {3.45386}, {3.46785}, {3.48183}, {3.49581}, {3.5098}, \ {3.52378}, {3.53776}, {3.55175}, {3.56573}, {3.57971}, {3.5937}, \ {3.60768}, {3.62166}, {3.63565}, {3.64963}, {3.66361}, {3.6776}, \ {3.69158}, {3.70556}, {3.71955}, {3.73353}, {3.74751}, {3.7615}, \ {3.77548}, {3.78946}, {3.80345}, {3.81743}, {3.83141}, {3.8454}, \ {3.85938}, {3.87336}, {3.88735}, {3.90133}, {3.91531}, {3.9293}, \ {3.94328}, {3.95726}, {3.97124}, {3.98523}, {3.99921}, {4.01319}, \ {4.02718}, {4.04116}, {4.05514}, {4.06913}, {4.08311}, {4.09709}, \ {4.11108}, {4.12506}, {4.13904}, {4.15303}, {4.16701}, {4.18099}, \ {4.19498}, {4.20896}, {4.22294}, {4.23693}, {4.25091}, {4.26489}, \ {4.27888}, {4.29286}, {4.30684}, {4.32083}, {4.33481}, {4.34879}, \ {4.36278}, {4.37676}, {4.39074}, {4.40473}, {4.41871}, {4.43269}, \ {4.44668}, {4.46066}, {4.47464}, {4.48863}, {4.50261}, {4.51659}, \ {4.53057}, {4.54456}, {4.55854}, {4.57253}, {4.58651}, {4.60049}, \ {4.61447}, {4.62846}, {4.64244}, {4.65642}, {4.67041}, {4.68439}, \ {4.69837}, {4.71236}, {4.72634}, {4.74032}, {4.75431}, {4.76829}, \ {4.78227}, {4.79626}, {4.81024}, {4.82422}, {4.83821}, {4.85219}, \ {4.86617}, {4.88016}, {4.89414}, {4.90812}, {4.92211}, {4.93609}, \ {4.95007}, {4.96406}, {4.97804}, {4.99202}, {5.00601}, {5.01999}, \ {5.03397}, {5.04796}, {5.06194}, {5.07592}, {5.08991}, {5.10389}, \ {5.11787}, {5.13186}, {5.14584}, {5.15982}, {5.17381}, {5.18779}, \ {5.20177}, {5.21575}, {5.22974}, {5.24372}, {5.2577}, {5.27169}, \ {5.28567}, {5.29965}, {5.31364}, {5.32762}, {5.3416}, {5.35559}, \ {5.36957}, {5.38355}, {5.39754}, {5.41152}, {5.4255}, {5.43949}, \ {5.45347}, {5.46745}, {5.48144}, {5.49542}, {5.5094}, {5.52339}, \ {5.53737}, {5.55135}, {5.56534}, {5.57932}}

epsilon2[w_]:=2/Pi NIntegrate[a epsilon1[a]/(a^2-w^2), {a, 0, 5.5793}, Method-> "PrincipalValue", MaxRecursion->20, Exclusions->{(a^2-w^2)==0}]//Quiet

Plot[epsilon2[w],{w,0,5.57932},AxesOrigin->{0,0},PlotPoints->400]

However, by using the code above, I managed only to get a list of irrelevant numbers. The graph of epsilon1 looks like a Sine curve, while that of epsilon2 looks like a "M" shape. Any help is greatly appreciated! =) Thanks!