"A single transition."

STEP(t-x)

f1(t,x):=STEP(t-x)-STEP(t-2+x)-STEP(t-2-x)+STEP(t-4+x)

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

FOURIER(f1(t,x),t,0,4,15)

x:epsilonReal (0, 1)

;Simp(#5)
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(t):=0.53*SIN(pi*t/2)

f(t)

;Int(#14,t)
INT(f(t),t,0,x)

f1(x):=53/(50*pi)-53*COS(pi*x/2)/(50*pi)

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

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)

test:=VECTOR([n,SUM(-(-1)^i*4*COS(n*pi*xv SUB i/2),i,1,DIM(xv))/(n*pi)],n,1,3~
1,2)

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(#29')
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(x)-fp1(x,v)

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

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

sx([],0.02)

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

xv:=fe(0.02,10)

f1(x)-fp1(x,xv)

PrecisionDigits:=10

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)

xv:=fe(magic SUB 13 SUB 1 SUB 1,magic SUB 13 SUB 1 SUB 2)

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

VECTOR([x SUB 1 SUB 2,t5(5,x SUB 2)],x,magic)

xv:=fe(magic SUB 1 SUB 1 SUB 1,magic SUB 1 SUB 1 SUB 2)

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

magic1:=REPLACE([[0.008965773545,magic SUB 1 SUB 1 SUB 2],[0.1470880587,0.167~
6917903,0.2685116118,0.2916705834,0.360715589,0.3860901681,0.4419862194,0.469~
4799812,0.5178231964,0.5473851564,0.5907951619,0.6223609381,0.6623450627,0.69~
58004882,0.7333496337,0.7685018328,0.8043372825,0.8408941448,0.8755880199,0.9~
131524318,0.9471941345,0.9852795819]],magic,1)

magic3:=APPEND(magic2,[[[0.02473554187,8],xv]])

VECTOR([x SUB 1 SUB 2,t5(5,x SUB 2)],x,magic)

t5(3,xv)

magic1:=APPEND(magic,[[[0.03300657189,6],xv]])

;Approx(#52')
magic1:=[[[0.008965773545,22],[0.1470880587,0.1676917901,0.2685116118,0.29167~
05833,0.360715589,0.386090168,0.4419862193,0.4694799812,0.5178231963,0.547385~
1563,0.5907951618,0.6223609381,0.6623450627,0.6958004882,0.7333496337,0.76850~
18325,0.8043372822,0.8408941445,0.8755880199,0.9131524318,0.9471941345,0.9852~
795817]],[[0.009335,21],[0.1501000016,0.1716214485,0.2743827147,0.2986466287,~
0.3690379151,0.3956906182,0.4527009849,0.4816431875,0.5309863246,0.562161013,~
0.6065181768,0.6398447122,0.6807657956,0.7160974536,0.7546092333,0.7916997841~
,0.8285583284,0.8670410305,0.9028553633,0.9422434392,0.9775443959]],[[0.00986~
5,20],[0.1543225993,0.1771695206,0.2826460976,0.30851522,0.3807933644,0.40931~
14664,0.4678892321,0.4989524955,0.549710148,0.5832498491,0.6289578176,0.66486~
25463,0.7071343011,0.7451973613,0.7851152485,0.8249949354,0.8633707308,0.9045~
719501,0.9420754419,0.983965987]],[[0.01031,19],[0.1577824077,0.1817495801,0.~
2894454508,0.3166797109,0.390503749,0.4206160314,0.4804833409,0.5133653939,0.~
5652945225,0.6008642687,0.6477015448,0.6858135224,0.7292285261,0.7696115272,0~
.8107375998,0.8529505814,0.8926523865,0.9360729841,0.9750796608]],[[0.010964,~
18],[0.1627364099,0.1883620249,0.2992272745,0.3284965123,0.4045346266,0.43703~
60455,0.4987596644,0.5343768346,0.5880059198,0.626629587,0.6751245122,0.71654~
22937,0.7616606466,0.8054787009,0.8484373399,0.8940337296,0.9357871294,0.9823~
222909]],[[0.01151,17],[0.1667620287,0.1937830934,0.3072167686,0.3382110196,0~
.4160494362,0.4505875163,0.5138295547,0.5517860112,0.6068188511,0.648053019,0~
.6979352949,0.7421602019,0.7887290479,0.8354191325,0.8799703734,0.9283192857,~
0.971895483]],[[0.01233724501,16],[0.1726865452,0.2018405153,0.3190432198,0.3~
526961638,0.4331873533,0.4708838768,0.5363799551,0.5779753752,0.6351167759,0.~
6804031321,0.7324043916,0.7809414497,0.8297731616,0.8807772759,0.9278743907,0~
.9802030208]],[[0.01302904506,15],[0.1774928827,0.208447629,0.328698892,0.364~
6168915,0.4472643252,0.4876691206,0.5550137335,0.599737414,0.6586327743,0.707~
3878671,0.761187159,0.8133602438,0.8641601677,0.9186922389,0.9680659227]],[[0~
.01410693979,14],[0.1847389229,0.2185301562,0.3433629777,0.3828856867,0.46879~
38145,0.5135390028,0.5837110548,0.6334560767,0.6950835046,0.7493616087,0.8060~
330821,0.8638615272,0.9179031388,0.9776814443]],[[0.01500616876,13],[0.190578~
9462,0.2267648451,0.3552784744,0.3978787085,0.4864263034,0.5349036664,0.60739~
68134,0.6614592064,0.7253799968,0.7843451241,0.8435015588,0.9059706213,0.9629~
182198]],[[0.01646476583,12],[0.1996994953,0.2398239301,0.3740671398,0.421795~
1445,0.5144922267,0.5692342937,0.6454452446,0.7067307056,0.7744349579,0.84106~
49855,0.9044840867,0.9741512787]],[[0.01768855683,11],[0.2070518633,0.2505321~
527,0.3893794565,0.4415392854,0.5376125677,0.5978063427,0.6771149747,0.744635~
1828,0.8156103995,0.8886254934,0.9559067781]],[[0.0197712181,10],[0.219016937~
4,0.2683150203,0.4146322165,0.4746052041,0.5762516365,0.6461104212,0.73070582~
06,0.809075996,0.8859240259,0.9693828682]],[[0.02153364626,9],[0.228672095,0.~
2829977685,0.4353276737,0.5021779012,0.6084136146,0.6867990361,0.7759435886,0~
.8635672869,0.9458028364]],[[0.02473554187,8],[0.2452829877,0.3089858606,0.47~
16453915,0.5516046585,0.6659985726,0.7605365586,0.8583185816,0.9622854615]],[~
[0.02750438124,7],[0.2588292503,0.330893348,0.5019816832,0.5939005402,0.71527~
9887,0.8242315764,0.9301059845]],[[0.03300657188,6],[0.2839376332,0.373311629~
7,0.5601085898,0.6773919461,0.8128942167,0.9503798616]]]

VECTOR([x SUB 1 SUB 2,t5(5,x SUB 2)],x,magic1)

f1(x)-fp1(x,xv)

VECTOR([x SUB 1 SUB 2,t5(3,x SUB 2)],x,magic1)

;Sub(#27')
test:=VECTOR([n,t5(n,xv)],n,1,31,2)

;User=Simp(User)
DIM(magic1)=17

xv:=magic1 SUB 17 SUB 2

t5(5,xv)

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

;Approx(#60)
-0.003977851598

;Sub(User')
xv:=fe(0.03214460505,6)

;Approx(#63')
xv:=[0.2801437442,0.3667448332,0.5511566108,0.6643355179,0.7975802877,0.93069~
33861]

;Approx(#60)
0.0002601988425