"A single transition."

STEP(t-x)

fp2(t,x):=STEP(t-x)-STEP(t-2+x)-STEP(t-2-x)+STEP(t-4+x)

FOURIER(y,t,t1,t2,n)

FOURIER(fp2(t,x),t,0,4,15)

x:epsilonReal (0, 1)

;Simp(User)
4*SIN(15*pi*t/2)*COS(15*pi*x/2)/(15*pi)+4*SIN(13*pi*t/2)*COS(13*pi*x/2)/(13*p~
i)+4*SIN(11*pi*t/2)*COS(11*pi*x/2)/(11*pi)+4*SIN(9*pi*t/2)*COS(9*pi*x/2)/(9*p~
i)+4*SIN(7*pi*t/2)*COS(7*pi*x/2)/(7*pi)+4*SIN(5*pi*t/2)*COS(5*pi*x/2)/(5*pi)+~
4*SIN(3*pi*t/2)*COS(3*pi*x/2)/(3*pi)+4*SIN(pi*t/2)*COS(pi*x/2)/pi

cx=COS(pi*x/2)

CHEBYCHEV_T(n,x)

CHEBYCHEV_T(n,COS(pi*x/2))=COS(n*pi*x/2)

NEWTONS(u,x,x0,n)

pd:=[17.9125,21.4007,36.1121,42.7902,54.8818,64.1028,74.4503,85.1345]

f(amp,t):=amp*SIN(pi*t/2)

f1(amp,x):=INT(f(amp,t),t,0,x)

;Simp(#14')
f1(amp,x):=2*amp/pi-2*amp*COS(pi*x/2)/pi

cva:=VECTOR(SUM((-1)^(i+1)*CHEBYCHEV_T(n,cvx SUB i),i,1,DIM(cvx)),n,1,DIM(cvx~
)*2-1,2)

VECTOR(SUM((-1)^(i+1)*CHEBYCHEV_T(n,cvx SUB i),i,1,DIM(cvx)),n,3,2*DIM(cvx)+1~
,2)

cvb:=VECTOR(IF(i=1,cva SUB i-0.53*pi/4,cva SUB i),i,1,DIM(cva))

cvx0:=VECTOR(COS(pi*x/2),x,xv)

cvx1:=NEWTONS(cvb,cvx,cvx0,3)

cvx1:=(NEWTONS(cvb,cvx,cvx0,10)) SUB 10

p1:=VECTOR(180*ACOS(x)/pi,x,cvx1)

p0:=VECTOR(180*ACOS(x)/pi,x,cvx0)

cvx1:=NEWTONS(cvb,cvx,cvx0,4)

SUM(4*SIN(n*pi*t/2)*COS(n*pi*x/2)/(n*pi),n,1,15,2)

VECTOR(4*COS(n*pi*x/2)/(n*pi),n,1,15,2)

fp(t,v):=SUM(-(-1)^i*STEP(t-v SUB i),i,1,DIM(v))

fp1(x,v):=SUM(-(-1)^i*INT(STEP(t-v SUB i),t,0,x),i,1,DIM(v))

;Simp(#28')
fp1(x,v):=SUM(-(-1)^i*((v SUB i-x)^2/(2*ABS(v SUB i-x))-v SUB i^2/(2*ABS(v SU~
B i))+x/2),i,1,DIM(v))

f1(amp,x)-fp1(x,v)

NSOLUTIONS(f1(amp,x)-fp1(x,[])=me,x,0,1)

sx(amp,v,e):=NSOLUTIONS(f1(amp,x)-fp1(x,v)-(-1)^DIM(v)*e,x,MAX(0,MAX(v)),1)

fe(amp,e,n):=ITERATE(APPEND(v,sx(amp,v,e)),v,[],n)

p0:=VECTOR(90*x,x,xv)

POLY_INTERPOLATE([[x1,y1],[x2,y2]],x)

;Simp(Solve(User,x))
x=(x1*y2-x2*y1)/(y2-y1)

(x1*y2-x2*y1)/(y2-y1)

t5(n,v):=SUM(-(-1)^i*4*COS(n*pi*v SUB i/2),i,1,DIM(v))/(n*pi)

test:=VECTOR([n,t5(n,xv)],n,1,63,2)

PrecisionDigits:=8

xv:=fe(1.0204,0.00671,19)

;Approx(#41')
xv:=[0.091581052,0.10753678,0.16866527,0.1873457,0.22835909,0.2498053,0.28221~
839,0.30686069,0.33412408,0.36273606,0.38643302,0.42032878,0.44133016,0.48290~
721,0.50173568,0.55614375,0.57311143,0.6562183,0.67144424]

;Approx(User)
[4.4695007*10^(-6),-5.4419099*10^(-5)]

[[[0.923475,0.078764,1],[1/3]],[[1.03982,0.038493,3],[0.218169,0.359352,0.480~
366]],[[1.029,0.0237671,5],[0.172011,0.242877,0.346449,0.469721,0.534845]],[[~
1.0268,0.0173406,7],[0.146963,0.194609,0.286278,0.354999,0.414463,0.526632,0.~
571127]],[[1.02543,0.0136871,9],[0.130592,0.166472,0.24968,0.297463,0.352144,~
0.41774,0.459729,0.564091,0.59784]],[[1.024,0.0113138,11],[0.118779,0.147515,~
0.224315,0.260856,0.311704,0.358282,0.397781,0.460439,0.492904,0.591052,0.618~
224]],[[1.023,0.00965615,13],[0.109763,0.133737,0.205478,0.235048,0.282781,0.~
318919,0.356249,0.401521,0.432512,0.492757,0.519209,0.61279,0.635516]],[[1.02~
2,0.0084211433,15],[0.10253796,0.12308097,0.19067091,0.21545607,0.26058666,0.~
29004633,0.32550354,0.36093207,0.39054605,0.43449028,0.45999651,0.51804248,0.~
54035027,0.62987713,0.64940307]],[[1.021,0.0074704271,17],[0.096612148,0.1145~
823,0.17870326,0.20002375,0.24295344,0.26779881,0.30163955,0.33073339,0.35911~
28,0.39378626,0.41836226,0.46109332,0.48276564,0.5389144,0.55819694,0.6443476~
3,0.66146309]],[[1.0204,0.00671,19],[0.091581052,0.10753678,0.16866527,0.1873~
457,0.22835909,0.2498053,0.28221839,0.30686069,0.33412408,0.36273606,0.386433~
02,0.42032878,0.44133016,0.48290721,0.50173568,0.55614375,0.57311143,0.656218~
3,0.67144424]],[[1.019534,0.006097,21],[0.087327866,0.10168715,0.16025621,0.1~
7688772,0.2162518,0.23512501,0.26629799,0.28768003,0.31396044,0.3383357,0.361~
24011,0.38938844,0.40975634,0.4429668,0.46131156,0.50191515,0.51856451,0.5715~
5308,0.58670626,0.66745533,0.68117211]],[[1.019,0.005585,23],[0.083597182,0.0~
9664268,0.15294141,0.16791608,0.20580876,0.22264256,0.25270042,0.27156254,0.2~
9694924,0.31815815,0.3403299,0.36438284,0.38414835,0.41181678,0.42967521,0.46~
221896,0.4784969,0.51818344,0.5330976,0.584775,0.59845679,0.67706235,0.689536~
19]],[[1.0183349,0.0051528341,25],[0.080319484,0.092269639,0.14655653,0.16016~
943,0.19675288,0.2119368,0.24099692,0.25785949,0.28244056,0.30119864,0.322702~
75,0.34368995,0.36289748,0.38661028,0.40400512,0.43120143,0.44710403,0.479012~
9,0.49364276,0.53246949,0.54597578,0.59642738,0.60889893,0.68545468,0.6968948~
]]]

se(v):=SUM((t5(n,v)/n)^2,n,3,63,2)

se(xv)

(x1*y2-x2*y1)/(y2-y1)

[t5(3,xv),t5(5,xv)]

f1(1.017,x)-fp1(x,xv)

f1(1.0183349,x)-fp1(x,xv)

;Sub(User')
xv:=fe(1.0183349,0.0051528341,25)

;Approx(#51')
xv:=[0.080319484,0.092269639,0.14655653,0.16016943,0.19675288,0.21193680,0.24~
099692,0.25785949,0.28244056,0.30119864,0.32270275,0.34368995,0.36289748,0.38~
661028,0.40400512,0.43120143,0.44710403,0.47901290,0.49364276,0.53246949,0.54~
597578,0.59642738,0.60889893,0.68545468,0.69689480]

;Approx(#48)
[6.1369516*10^(-5),1.7065062*10^(-6)]