(* First we write the fitted values of F_L for various ranges of w', which we call x here. *)
FLwp01top05 = (-2.074368068576991`+0.8155119939540679` I)-((0.` +0.2500336106590544` I)/x)+(204.16372960015224` +8.409910464019896` I) x-(16854.40538745633` +375.4007016267882` I) (x^2)+(1.0824815356393207`*^6+12987.875896829495` I) (x^3)-(4.951683127747867`*^7+382955.42295639653` I) (x^4)+(1.598764789237864`*^9+8.534176596745986`*^6 I) (x^5)-(3.610248568745412`*^10+1.3671691697790703`*^8 I) (x^6)+(5.57081762319418`*^11+1.505865548423325`*^9 I) (x^7)-(5.590893388948593`*^12+1.0654633685349174`*^10 I) (x^8)+(3.2860592394589914`*^13+4.259609867339026`*^10 I) (x^9)-(8.577298343946184`*^13+7.004867548143304`*^10 I) (x^10);

FLwp01top05s = (-2.074368068576991`-0.8155119939540679` I)+((0.` +0.2500336106590544` I)/x)+(204.16372960015224` -8.409910464019896` I) x-(16854.40538745633` -375.4007016267882` I) (x^2)+(1.0824815356393207`*^6-12987.875896829495` I) (x^3)-(4.951683127747867`*^7-382955.42295639653` I) (x^4)+(1.598764789237864`*^9-8.534176596745986`*^6 I) (x^5)-(3.610248568745412`*^10-1.3671691697790703`*^8 I) (x^6)+(5.57081762319418`*^11-1.505865548423325`*^9 I) (x^7)-(5.590893388948593`*^12-1.0654633685349174`*^10 I) (x^8)+(3.2860592394589914`*^13-4.259609867339026`*^10 I) (x^9)-(8.577298343946184`*^13-7.004867548143304`*^10 I) (x^10);

FLwp05top1=(-1.546182274429809`-38.06984017013995` I)+(76.86552013593213` +2740.8625814904162` I) x-(2280.588071704044` +114877.88430687401` I) (x^2)+(55806.01001771269` +3.1915663051057965`*^6 I) (x^3)-(1.0113942390972467`*^6+6.1735418144253395`*^7 I) (x^4)+(1.3333882672060218`*^7+8.48227368861889`*^8 I) (x^5)-(1.2645704495701769`*^8+8.278078970168419`*^9 I) (x^6)+(8.413609409366856`*^8+5.623810841245506`*^10 I) (x^7)-(3.7303106992945695`*^9+2.533105084768257`*^11 I) (x^8)+(9.904152250143179`*^9+6.808805104755027`*^11 I) (x^9)-(1.1918862823713745`*^10+8.274553918074298`*^11 I) (x^10);

FLwp05top1s = (-1.546182274429809`+38.06984017013995` I)+(76.86552013593213` -2740.8625814904162` I) x-(2280.588071704044` -114877.88430687401` I) (x^2)+(55806.01001771269` -3.1915663051057965`*^6 I) (x^3)-(1.0113942390972467`*^6-6.1735418144253395`*^7 I) (x^4)+(1.3333882672060218`*^7-8.48227368861889`*^8 I) (x^5)-(1.2645704495701769`*^8-8.278078970168419`*^9 I) (x^6)+(8.413609409366856`*^8-5.623810841245506`*^10 I) (x^7)-(3.7303106992945695`*^9-2.533105084768257`*^11 I) (x^8)+(9.904152250143179`*^9-6.808805104755027`*^11 I) (x^9)-(1.1918862823713745`*^10-8.274553918074298`*^11 I) (x^10);

FLwp1top15 = (-1.336841209205315`-22.842066478021` I)+(52.59400705658776` +1021.2570362736781` I) x-(1007.7222570865857` +26214.79549984712` I) (x^2)+(16013.926767144563` +446596.5879327631` I) (x^3)-(189953.12325537266` +5.3137463471869435`*^6 I) (x^4)+(1.6348689809870988`*^6+4.5036575179860584`*^7 I) (x^5)-(1.0088391905852713`*^7+2.718313771435049`*^8 I) (x^6)+(4.358158546582263`*^7+1.1449944231086628`*^9 I) (x^7)-(1.253089836236247`*^8+3.205451380978863`*^9 I) (x^8)+(2.1559585404296193`*^8+5.36795241549547`*^9 I) (x^9)-(1.6803872291463724`*^8+4.07379060740384`*^9 I) (x^10);

FLwp1top15s =(-1.336841209205315`+22.842066478021` I)+(52.59400705658776` -1021.2570362736781` I) x-(1007.7222570865857` -26214.79549984712` I) (x^2)+(16013.926767144563` -446596.5879327631` I) (x^3)-(189953.12325537266` -5.3137463471869435`*^6 I) (x^4)+(1.6348689809870988`*^6-4.5036575179860584`*^7 I) (x^5)-(1.0088391905852713`*^7-2.718313771435049`*^8 I) (x^6)+(4.358158546582263`*^7-1.1449944231086628`*^9 I) (x^7)-(1.253089836236247`*^8-3.205451380978863`*^9 I) (x^8)+(2.1559585404296193`*^8-5.36795241549547`*^9 I) (x^9)-(1.6803872291463724`*^8-4.07379060740384`*^9 I) (x^10);

FLwp15top25 =(-1.054030749802681`-13.38772854007396` I)+(32.50931642174401` +371.3467483878254` I) x-(354.2463818939257` +5760.21216335419` I) (x^2)+(3206.113509211104` +58927.463904026976` I) (x^3)-(22798.713102417154` +421315.6589137102` I) (x^4)+(120029.9509999186` +2.1490827806768385`*^6 I) (x^5)-(453570.85149529483` +7.815093963208258`*^6 I) (x^6)+(1.195325339729558`*^6+1.9844326953115996`*^7 I) (x^7)-(2.0895047041714564`*^6+3.34998960696377`*^7 I) (x^8)+(2.1815066993926624`*^6+3.383318747010579`*^7 I) (x^9)-(1.0311762160306388`*^6+1.5486156389718438`*^7 I) (x^10);

FLwp15top25s =  (-1.054030749802681`+13.38772854007396` I)+(32.50931642174401` -371.3467483878254` I) x-(354.2463818939257` -5760.21216335419` I) (x^2)+(3206.113509211104` -58927.463904026976` I) (x^3)-(22798.713102417154` -421315.6589137102` I) (x^4)+(120029.9509999186` -2.1490827806768385`*^6 I) (x^5)-(453570.85149529483` -7.815093963208258`*^6 I) (x^6)+(1.195325339729558`*^6-1.9844326953115996`*^7 I) (x^7)-(2.0895047041714564`*^6-3.34998960696377`*^7 I) (x^8)+(2.1815066993926624`*^6-3.383318747010579`*^7 I) (x^9)-(1.0311762160306388`*^6-1.5486156389718438`*^7 I) (x^10);

FLwp25top5 =(-0.765315365618325`-6.88323316335463` I)+(20.215776205341697` +111.65280426252659` I) x-(114.86196123926942` +974.3580455881528` I) (x^2)+(400.43251061062506` +5433.366347720079` I) (x^3)-(900.7710476850195` +20784.890911081195` I) (x^4)+(1257.371859344108` +56148.62437267807` I) (x^5)-(840.5334049554463` +107514.6753184464` I) (x^6)-(347.7151844786713` -143242.7405463029` I) (x^7)+(1227.5649995954782` -126554.30260652385` I) (x^8)-(995.1713074562813` -66750.86229021192` I) (x^9)+(297.3010086695773` -15925.534414923093` I) (x^10);

FLwp25top5s =(-0.765315365618325`+6.88323316335463` I)+(20.215776205341697` -111.65280426252659` I) x-(114.86196123926942` -974.3580455881528` I) (x^2)+(400.43251061062506` -5433.366347720079` I) (x^3)-(900.7710476850195` -20784.890911081195` I) (x^4)+(1257.371859344108` -56148.62437267807` I) (x^5)-(840.5334049554463` -107514.6753184464` I) (x^6)-(347.7151844786713` +143242.7405463029` I) (x^7)+(1227.5649995954782` +126554.30260652385` I) (x^8)-(995.1713074562813` +66750.86229021192` I) (x^9)+(297.3010086695773` +15925.534414923093` I) (x^10);

FLwp5to1 =(-0.22369023113482112`-2.4230738617506526` I)+(11.659467998348347` +19.18157775493822` I) x-(55.282603699201914` +86.77147947851805` I) (x^2)+(163.85177156072228` +254.25511241275098` I) (x^3)-(329.22201419244595` +511.3289762267318` I) (x^4)+(463.33413802346354` +723.6878390686252` I) (x^5)-(459.42672925812116` +722.8871731885515` I) (x^6)+(315.5288178996529` +500.3580100635709` I) (x^7)-(143.20717192728142` +228.84023156006018` I) (x^8)+(38.70178145134517` +62.29473126072145` I) (x^9)-(4.721636404464535` +7.651538621550049` I) (x^10);

FLwp5to1s =(-0.22369023113482112`+2.4230738617506526` I)+(11.659467998348347` -19.18157775493822` I) x-(55.282603699201914` -86.77147947851805` I) (x^2)+(163.85177156072228` -254.25511241275098` I) (x^3)-(329.22201419244595` -511.3289762267318` I) (x^4)+(463.33413802346354` -723.6878390686252` I) (x^5)-(459.42672925812116` -722.8871731885515` I) (x^6)+(315.5288178996529` -500.3580100635709` I) (x^7)-(143.20717192728142` -228.84023156006018` I) (x^8)+(38.70178145134517` -62.29473126072145` I) (x^9)-(4.721636404464535` -7.651538621550049` I) (x^10);

FLw1to2 =(0.7251370811397575` -0.9956012068957016` I)+(1.3030573869593893` +3.5406609137916787` I) x-(3.0971015157780446` +7.473327325227952` I) (x^2)+(4.6157716161738245` +10.427335404554817` I) (x^3)-(4.6689580691564405` +10.1092656903019` I) (x^4)+(3.3088944288870006` +6.953092492527102` I) (x^5)-(1.6519336832113014` +3.3939370734743894` I) (x^6)+(0.5710104204063646` +1.1524742903176524` I) (x^7)-(0.13037712556021266` +0.2593311984685234` I) (x^8)+(0.017717039705709238` +0.034808655227718324` I) (x^9)-(0.0010863537021469992` +0.0021116299070783485` I) (x^10);

FLw1to2s =(0.7251370811397575` +0.9956012068957016` I)+(1.3030573869593893` -3.5406609137916787` I) x-(3.0971015157780446` -7.473327325227952` I) (x^2)+(4.6157716161738245` -10.427335404554817` I) (x^3)-(4.6689580691564405` -10.1092656903019` I) (x^4)+(3.3088944288870006` -6.953092492527102` I) (x^5)-(1.6519336832113014` -3.3939370734743894` I) (x^6)+(0.5710104204063646` -1.1524742903176524` I) (x^7)-(0.13037712556021266` -0.2593311984685234` I) (x^8)+(0.017717039705709238` -0.034808655227718324` I) (x^9)-(0.0010863537021469992` -0.0021116299070783485` I) (x^10);

FLw2to3=(0.9419854051121342` +0.0897648649325613` I)+(0.11617143067028024` -1.5516166378308358` I) x-(0.10050938355216658` -3.6091781973051855` I) (x^2)+(0.034416338737192256` -4.357791155831771` I) (x^3)+(0.011589670373896934` +3.2997351435232165` I) (x^4)-(0.018412441620718215` +1.675937986744152` I) (x^5)+(0.009502487268614386` +0.5834410767404163` I) (x^6)-(0.002795466504494883` +0.13804796234121774` I) (x^7)+(0.0004967306939672501` +0.021294034640919404` I) (x^8)-(0.00004992540916547485` +0.0019361467567940794` I) (x^9)+(2.192055057315053`*^-6+0.00007886529391123477` I) (x^10);

FLw2to3s=(0.9419854051121342` -0.0897648649325613` I)+(0.11617143067028024` +1.5516166378308358` I) x-(0.10050938355216658` +3.6091781973051855` I) (x^2)+(0.034416338737192256` +4.357791155831771` I) (x^3)+(0.011589670373896934` -3.2997351435232165` I) (x^4)-(0.018412441620718215` -1.675937986744152` I) (x^5)+(0.009502487268614386` -0.5834410767404163` I) (x^6)-(0.002795466504494883` -0.13804796234121774` I) (x^7)+(0.0004967306939672501` -0.021294034640919404` I) (x^8)-(0.00004992540916547485` -0.0019361467567940794` I) (x^9)+(2.192055057315053`*^-6-0.00007886529391123477` I) (x^10);

FLw3to5=(0.9808264981912109` +0.11142176156646857` I)+(0.019654912350643837` -0.7452029846991235` I) x-(0.0055558517209833205` -1.0371463542334993` I) (x^2)-(0.0032075175173276963` +0.7674790203635579` I) (x^3)+(0.003502565171043913` +0.358355819114527` I) (x^4)-(0.0015161258502251027` +0.11246463987983421` I) (x^5)+(0.00039027087032397865` +0.024206704045560475` I) (x^6)-(0.00006409002059397282` +0.00354102653060151` I) (x^7)+(6.62709726880779`*^-6+0.00033756475249341626` I) (x^8)-(3.953370377028219`*^-7+0.000018958299261538012` I) (x^9)+(1.0413637217533757`*^-8+4.76678835974586`*^-7 I) (x^10);

FLw3to5s = (0.9808264981912109` -0.11142176156646857` I)+(0.019654912350643837` +0.7452029846991235` I) x-(0.0055558517209833205` +1.0371463542334993` I) (x^2)-(0.0032075175173276963` -0.7674790203635579` I) (x^3)+(0.003502565171043913` -0.358355819114527` I) (x^4)-(0.0015161258502251027` -0.11246463987983421` I) (x^5)+(0.00039027087032397865` -0.024206704045560475` I) (x^6)-(0.00006409002059397282` -0.00354102653060151` I) (x^7)+(6.62709726880779`*^-6-0.00033756475249341626` I) (x^8)-(3.953370377028219`*^-7-0.000018958299261538012` I) (x^9)+(1.0413637217533757`*^-8-4.76678835974586`*^-7 I) (x^10);

FLw5to7 =(0.9981120785908549` +0.14348960825434376` I)-(0.002455773987275575` +0.33556564792298593` I) x+(0.0029383272089226276` +0.22377341632805786` I) (x^2)-(0.0012134741707197653` +0.07450632716274133` I) (x^3)+(0.00023660716713946438` +0.012703214108506689` I) (x^4)-(0.000012709619867469885` +0.0005200709623713266` I) (x^5)-(4.086320014566328`*^-6+0.0002291720363358329` I) (x^6)+(1.0171856234296336`*^-6+0.000051981923403471365` I) (x^7)-(1.0604444062223649`*^-7+5.200447803831324`*^-6 I) (x^8)+(5.5872182218940835`*^-9+2.666419952124441`*^-7 I) (x^9)-(1.2179718380500506`*^-10+5.6964538014303815`*^-9 I) (x^10);

FLw5to7s =(0.9981120785908549` -0.14348960825434376` I)-(0.002455773987275575` -0.33556564792298593` I) x+(0.0029383272089226276` -0.22377341632805786` I) (x^2)-(0.0012134741707197653` -0.07450632716274133` I) (x^3)+(0.00023660716713946438` -0.012703214108506689` I) (x^4)-(0.000012709619867469885` -0.0005200709623713266` I) (x^5)-(4.086320014566328`*^-6-0.0002291720363358329` I) (x^6)+(1.0171856234296336`*^-6-0.000051981923403471365` I) (x^7)-(1.0604444062223649`*^-7-5.200447803831324`*^-6 I) (x^8)+(5.5872182218940835`*^-9-2.666419952124441`*^-7 I) (x^9)-(1.2179718380500506`*^-10-5.6964538014303815`*^-9 I) (x^10);

FLw7to10 = (0.9842976759066336` -0.8375590347811586` I)+(0.012677535955284074` +0.700500632565088` I) x-(0.004274982557942156` +0.2453353610357683` I) (x^2)+(0.0006763152981619956` +0.04009460271643188` I) (x^3)-(0.00002811701496468489` +0.0017677946087409024` I) (x^4)-(5.36047129571034`*^-6+0.0003187129827877537` I) (x^5)+(4.325966796740375`*^-7+0.00002707131082476127` I) (x^6)+(5.4805960630766876`*^-8+3.2918872720653907`*^-6 I) (x^7)-(4.350538866690276`*^-9+2.748777105058884`*^-7 I) (x^8)-(6.605640079472755`*^-10+4.02240910035073`*^-8 I) (x^9)+(3.921531984743002`*^-11+2.5018533303401927`*^-9 I) (x^10)+(7.967712705620964`*^-12+4.899648976531013`*^-10 I) (x^11)-(4.674747558075541`*^-13+2.970543318462095`*^-11 I) (x^12)-(8.715776285883256`*^-14+5.378031927378276`*^-12 I) (x^13)+(1.1764094313300902`*^-14+7.352577108368047`*^-13 I) (x^14)-(5.474492475768983`*^-16+3.4422007635526065`*^-14 I) (x^15)+(9.337841094642243`*^-18+5.896874703097634`*^-16 I) (x^16);

FLw7to10s = (0.9842976759066336` +0.8375590347811586` I)+(0.012677535955284074` -0.700500632565088` I) x-(0.004274982557942156` -0.2453353610357683` I) (x^2)+(0.0006763152981619956` -0.04009460271643188` I) (x^3)-(0.00002811701496468489` -0.0017677946087409024` I) (x^4)-(5.36047129571034`*^-6-0.0003187129827877537` I) (x^5)+(4.325966796740375`*^-7-0.00002707131082476127` I) (x^6)+(5.4805960630766876`*^-8-3.2918872720653907`*^-6 I) (x^7)-(4.350538866690276`*^-9-2.748777105058884`*^-7 I) (x^8)-(6.605640079472755`*^-10-4.02240910035073`*^-8 I) (x^9)+(3.921531984743002`*^-11-2.5018533303401927`*^-9 I) (x^10)+(7.967712705620964`*^-12-4.899648976531013`*^-10 I) (x^11)-(4.674747558075541`*^-13-2.970543318462095`*^-11 I) (x^12)-(8.715776285883256`*^-14-5.378031927378276`*^-12 I) (x^13)+(1.1764094313300902`*^-14-7.352577108368047`*^-13 I) (x^14)-(5.474492475768983`*^-16-3.4422007635526065`*^-14 I) (x^15)+(9.337841094642243`*^-18-5.896874703097634`*^-16 I) (x^16);

FLw10to100 = (0.9987529867005749` -0.06717756294610117` I)+(0.0004013273798942681` +0.016234404240302376` I) x-(0.0000651257773342993` +0.002344433356730908` I) (x^2)+(6.71372012599004`*^-6+0.00022682868059375184` I) (x^3)-(4.812566287154281`*^-7+0.000015625841855540188` I) (x^4)+(2.5158852813487544`*^-8+7.950068316380951`*^-7 I) (x^5)-(9.86622302361793`*^-10+3.057142642891729`*^-8 I) (x^6)+(2.952154965480685`*^-11+9.013629161906885`*^-10 I) (x^7)-(6.802715810891909`*^-13+2.0534349066385828`*^-11 I) (x^8)+(1.2107838327444977`*^-14+3.621864943975348`*^-13 I) (x^9)-(1.6593322395344873`*^-16+4.927480272040162`*^-15 I) (x^10)+(1.7335042863420503`*^-18+5.117015207057272`*^-17 I) (x^11)-(1.354344936347329`*^-20+3.9780006695810256`*^-19 I) (x^12)+(7.656134666208465`*^-23+2.2394267872159098`*^-21 I) (x^13)-(2.955983448749177`*^-25+8.615836489200894`*^-24 I) (x^14)+(6.970582788274622`*^-28+2.025603401481694`*^-26 I) (x^15)-(7.56999946908356`*^-31+2.1940693027651032`*^-29 I) (x^16);

FLw10to100s =  (0.9987529867005749` +0.06717756294610117` I)+(0.0004013273798942681` -0.016234404240302376` I) x-(0.0000651257773342993` -0.002344433356730908` I) (x^2)+(6.71372012599004`*^-6-0.00022682868059375184` I) (x^3)-(4.812566287154281`*^-7-0.000015625841855540188` I) (x^4)+(2.5158852813487544`*^-8-7.950068316380951`*^-7 I) (x^5)-(9.86622302361793`*^-10-3.057142642891729`*^-8 I) (x^6)+(2.952154965480685`*^-11-9.013629161906885`*^-10 I) (x^7)-(6.802715810891909`*^-13-2.0534349066385828`*^-11 I) (x^8)+(1.2107838327444977`*^-14-3.621864943975348`*^-13 I) (x^9)-(1.6593322395344873`*^-16-4.927480272040162`*^-15 I) (x^10)+(1.7335042863420503`*^-18-5.117015207057272`*^-17 I) (x^11)-(1.354344936347329`*^-20-3.9780006695810256`*^-19 I) (x^12)+(7.656134666208465`*^-23-2.2394267872159098`*^-21 I) (x^13)-(2.955983448749177`*^-25-8.615836489200894`*^-24 I) (x^14)+(6.970582788274622`*^-28-2.025603401481694`*^-26 I) (x^15)-(7.56999946908356`*^-31-2.1940693027651032`*^-29 I) (x^16);
(* Then, we split the integral over w' for the in modes into the various interval of w'. (FL is different in each interval)*)

IAHx0top01[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLwp01top05s*(x-w)),{x,0,1/100}];

IAHxp01top05P[v0_,v_,w_,ep_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLwp01top05s*(x-w)),{x,1/100,w,5/100},Method->{"PrincipalValue","SingularPointIntegrationRadius"->ep}];

IAHxp05top1[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLwp05top1s*(x-w)),{x,5/100,10/100}];

IAHxp1top15[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLwp1top15s*(x-w)),{x,10/100,15/100}];

IAHxp15top25[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLwp15top25s*(x-w)),{x,15/100,25/100}];

IAHxp25top5[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLwp25top5s*(x-w)),{x,25/100,50/100}];

IAHxp5to1[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLwp5to1s*(x-w)),{x,50/100,1}];

IAHx1to2[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLw1to2s*(x-w)),{x,1,2}];

IAHx2to3[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLw2to3s*(x-w)),{x,2,3}];

IAHx3to5[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLw3to5s*(x-w)),{x,3,5}];

IAHx5to7[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLw5to7s*(x-w)),{x,5,7}];

IAHx7to10[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLw7to10s*(x-w)),{x,7,10}];

IAHx10to100[v0_,v_,w_] := (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[I*x*(v0-v)]/(FLw10to100s*(x-w)),{x,10,100}];

IBHx0top01[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLwp01top05*(x+w)),{x,0,1/100}];

IBHxp01top05[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLwp01top05*(x+w)),{x,1/100,5/100}]

IBHxp05top1[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLwp05top1*(x+w)),{x,5/100,10/100}];

IBHxp1top15[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLwp1top15*(x+w)),{x,10/100,15/100}];

IBHxp15top25[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLwp15top25*(x+w)),{x,15/100,25/100}];

IBHxp25top5[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLwp25top5*(x+w)),{x,25/100,50/100}];

IBHxp5to1[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLwp5to1*(x+w)),{x,50/100,1}];

IBHx1to2[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLw1to2*(x+w)),{x,1,2}];

IBHx2to3[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLw2to3*(x+w)),{x,2,3}];

IBHx3to5[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLw3to5*(x+w)),{x,3,5}];

IBHx5to7[v0_,v_,w_]:= (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLw5to7*(x+w)),{x,5,7}];

IBHx7to10[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLw7to10*(x+w)),{x,7,10}];

IBHx10to100[v0_,v_,w_] := (-I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*NIntegrate[Exp[-I*x*(v0-v)]/(FLw10to100*(x+w)),{x,10,100}];
(* This is the contribution from Dirac delta when we split the integral for 0.01<w'<0.05 into a principal value integral and a *)

fanwp01top05[v_,w_] := Exp[-I*w*v]/(2*Sqrt[4*Pi*w]*(FLwp01top05s/.{x->w}))

K1 = Assuming[{v>v0>0,lam>0},Integrate[Sin[x*(v0-v)]/x,{x,lam,Infinity}]]

K2 = Assuming[{v>v0>0,lam>0},Integrate[Cos[x*(v0-v)]/x^2,{x,lam,Infinity}]]

(* This is the large x contribution to f^in *)

flam[v0_,v_,w_,lam_] = (I*Exp[-I*w*v0]/(2*Pi*Sqrt[4*Pi*w]))*(2*I*K1+2*(w-I/8)*K2)

(* Let v0=3 and Lambda=100. We want to plot the in-modes for w=0.02 on the future horizon. The first and second plots correspond to the real and imaginary parts of the in modes respectively. *)

L[j_,w_] := IAHx0top01[3,j,w]+IBHx0top01[3,j,w]+IAHxp01top05P[3,j,w]+IBHxp01top05[3,j,w]+IAHxp05top1[3,j,w]+IBHxp05top1[3,j,w]+IAHxp1top15[3,j,w]+IBHxp1top15[3,j,w]+IAHxp15top25[3,j,w]+IBHxp15top25[3,j,w]+IAHxp25top5[3,j,w]+IBHxp25top5[3,j,w]+IAHxp5to1[3,j,w]+IBHxp5to1[3,j,w]+IAHx1to2[3,j,w]+IBHx1to2[3,j,w]+IAHx2to3[3,j,w]+IBHx2to3[3,j,w]+IAHx3to5[3,j,w]+IBHx3to5[3,j,w]+IAHx5to7[3,j,w]+IBHx5to7[3,j,w]+IAHx7to10[3,j,w]+IBHx7to10[3,j,w]+IAHx10to100[3,j,w]+IBHx10to100[3,j,w];

finRelarge[j_,w_]:=flam[3,j,w,100] +L[j,w]+fanwp01top05[j,w]

pReal=Table[{j/50,Re[finRelarge[j/50,2/100]]},{j,151,1000}]

p1=ListPlot[pReal,Joined->True,PlotStyle->{Black,Thickness[0.003]}]

p2=Plot[Cos[(2/100)*3]/Sqrt[4*Pi*(2/100)],{x,0,3},PlotStyle->{Black,Thickness[0.003]}]

Show[p2,p1,PlotRange->{{0,20},{-1,3}},Frame->True,FrameStyle->Directive[Black,Thickness[0.003]],LabelStyle->{FontFamily->"Times" ,Black},Axes->False,FrameLabel->{"(v/M)",""}]

Im[finRelarge[3.01,2/100]]

pImaginary=Table[{j/50,Im[finRelarge[j/50,2/100]]},{j,151,1000}]

pIm={{3,-0.119861},{(151/50),-0.12010880638794263`},{(76/25),-0.12060381458849764`},{(153/50),-0.12109711322445481`},{(77/25),-0.12158883402810566`},{(31/10),-0.1220789326969222`},{(78/25),-0.12256737433010466`},{(157/50),-0.1230542173441181`},{(79/25),-0.12353944916877418`},{(159/50),-0.12402304362677913`},{(16/5),-0.12450503658569922`},{(161/50),-0.12498542729653613`},{(81/25),-0.12546419377506754`},{(163/50),-0.12594136084868515`},{(82/25),-0.1264169347177366`},{(33/10),-0.12689089590552025`},{(83/25),-0.12736326115721033`},{(167/50),-0.12783404139732216`},{(84/25),-0.1283032199516021`},{(169/50),-0.12877080714586883`},{(17/5),-0.1292368177053006`},{(171/50),-0.12970123866425576`},{(86/25),-0.13016407303895042`},{(173/50),-0.13062533503705892`},{(87/25),-0.13108501696143127`},{(7/2),-0.13154312034363122`},{(88/25),-0.13199965924716134`},{(177/50),-0.13245462814934006`},{(89/25),-0.13290802579298044`},{(179/50),-0.13335986602478106`},{(18/5),-0.1338101459336029`},{(181/50),-0.13425886149047125`},{(91/25),-0.1347060255913558`},{(183/50),-0.13515163865647356`},{(92/25),-0.13559569493113136`},{(37/10),-0.1360382055178211`},{(93/25),-0.13647917410228055`},{(187/50),-0.13691859436495515`},{(94/25),-0.13735647502192086`},{(189/50),-0.13779282237878587`},{(19/5),-0.13822763048807923`},{(191/50),-0.13866090544706833`},{(96/25),-0.13909265532948198`},{(193/50),-0.13952287523295737`},{(97/25),-0.13995156873021955`},{(39/10),-0.14037874484074078`},{(98/25),-0.1408044001534374`},{(197/50),-0.14122853604520852`},{(99/25),-0.14165116178446252`},{(199/50),-0.14207227570660183`},{4,-0.14249187745522632`},{(201/50),-0.14290997599229374`},{(101/25),-0.1433265714866126`},{(203/50),-0.14374166236056143`},{(102/25),-0.1441552568442119`},{(41/10),-0.14456735688319283`},{(103/25),-0.14497796018729595`},{(207/50),-0.14538707394650224`},{(104/25),-0.14579470170470543`},{(209/50),-0.14620084092081548`},{(21/5),-0.1466054975339837`},{(211/50),-0.14700867640961204`},{(106/25),-0.14741037515653765`},{(213/50),-0.1478105983600015`},{(107/25),-0.1482093518827451`},{(43/10),-0.148606633826851`},{(108/25),-0.1490024474209857`},{(217/50),-0.1493967991662826`},{(109/25),-0.1497896879175868`},{(219/50),-0.15018111562713132`},{(22/5),-0.15057108907650507`},{(221/50),-0.15095960806989583`},{(111/25),-0.15134667345873615`},{(223/50),-0.15173229197860238`},{(112/25),-0.1521164644996707`},{(9/2),-0.15249919099950823`},{(113/25),-0.15288047786981185`},{(227/50),-0.15326032708404957`},{(114/25),-0.15363873802397068`},{(229/50),-0.15401571650278909`},{(23/5),-0.15439126556193034`},{(231/50),-0.1547653842765998`},{(116/25),-0.15513807769273286`},{(233/50),-0.15550934980349862`},{(117/25),-0.15587919966444636`},{(47/10),-0.15624763143968926`},{(118/25),-0.15661464990558174`},{(237/50),-0.15698025436085725`},{(119/25),-0.15734444803038133`},{(239/50),-0.15770723624843222`},{(24/5),-0.15806861877819814`},{(241/50),-0.15842859792230501`},{(121/25),-0.15878717933144632`},{(243/50),-0.15914436340932256`},{(122/25),-0.15950015161178335`},{(49/10),-0.15985454964423534`},{(123/25),-0.16020755867290984`},{(247/50),-0.16055917944006687`},{(124/25),-0.16090941747858234`},{(249/50),-0.16125827476393095`},{5,-0.16160575151196485`},{(251/50),-0.16195185286559025`},{(126/25),-0.16229658161209218`},{(253/50),-0.16263993764947057`},{(127/25),-0.16298192556822`},{(51/10),-0.16332254891241996`},{(128/25),-0.16366180748638`},{(257/50),-0.16399970520880813`},{(129/25),-0.16433624626680882`},{(259/50),-0.16467143059005895`},{(26/5),-0.1650052613682788`},{(261/50),-0.16533774328220874`},{(131/25),-0.16566887658375623`},{(263/50),-0.1659986637248971`},{(132/25),-0.1663271096928331`},{(53/10),-0.166654215221669`},{(133/25),-0.1669799820756374`},{(267/50),-0.16730441535205404`},{(134/25),-0.16762751638187562`},{(269/50),-0.16794928633260375`},{(27/5),-0.16826973020842498`},{(271/50),-0.1685888499993765`},{(136/25),-0.16890664641192937`},{(273/50),-0.1692231241763047`},{(137/25),-0.16953828595247955`},{(11/2),-0.16985213214497893`},{(138/25),-0.17016466705268837`},{(277/50),-0.17047589396558518`},{(139/25),-0.17078581316717908`},{(279/50),-0.17109442841471612`},{(28/5),-0.1714017435451867`},{(281/50),-0.17170775889992326`},{(141/25),-0.17201247762581626`},{(283/50),-0.1723159039868327`},{(142/25),-0.17261803855501326`},{(57/10),-0.17291888385395573`},{(143/25),-0.173218444428847`},{(287/50),-0.17351672122620168`},{(144/25),-0.1738137161758842`},{(289/50),-0.17410943394241324`},{(29/5),-0.1744038759602559`},{(291/50),-0.17469704364317842`},{(146/25),-0.17498894160975315`},{(293/50),-0.17527957184796586`},{(147/25),-0.1755689353627213`},{(59/10),-0.17585703656955834`},{(148/25),-0.1761438780336705`},{(297/50),-0.176429460488203`},{(149/25),-0.17671378800839285`},{(299/50),-0.17699686370911843`},{6,-0.17727868819863882`},{(301/50),-0.17755926510520334`},{(151/25),-0.17783859802878618`},{(303/50),-0.1781166876073045`},{(152/25),-0.17839353695271512`},{(61/10),-0.17866915004617867`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p3=ListPlot[pImaginary,Joined->True,PlotStyle->{Black,Thickness[0.003]}]

p4=Plot[-Sin[(2/100)*3]/Sqrt[4*Pi*(2/100)],{x,0,3},PlotStyle->{Black,Thickness[0.003]}]

Show[p3,p4,PlotRange->{{0,20},{-0.3,0.1}},Frame->True,FrameStyle->Directive[Black,Thickness[0.003]],LabelStyle->{FontFamily->"Times" ,Black},Axes->False,FrameLabel->{"(v/M)",""}]