"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)

"Maximum error will be .03."

NSOLVE(f1(x)=0.03,x,0,1)

f2(x):=f1(x)-x+x1

NSOLVE(f2(x)=-0.03,x,0,1)

f3(x):=f1(x)-x2+x1

NSOLVE(f3(x)=0.03,x,0,1)

;Approx(#18)
x=0.2704888691365304641199530534425701294009927087905813561419952827

x1:=0.2704888691365304641199530534425701294009927087905813561419952827

;Approx(#20)
x=0.3502895366736790976225765560644814169883307013949708764145464157

x2:=0.3502895366736790976225765560644814169883307013949708764145464157

;Approx(#22)
x=0.5286525866798386923969486288210626828967126389278721116141522319

x3:=0.5286525866798386923969486288210626828967126389278721116141522319

f4(x):=f3(x)-x+x3

NSOLVE(f4(x)=-0.03,x,0,1)

;Approx(#30)
x=0.6318243510823024155058216486943852093174660250479437878101321588

x4:=0.6318243510823024155058216486943852093174660250479437878101321588

f5(x):=f3(x)-x4+x3

NSOLVE(f5(x)=0.03,x,0,1)

;Approx(#34)
x=0.7595378588469137714281901837807072184734684854060009528646602166

x5:=0.7595378588469137714281901837807072184734684854060009528646602166

f6(x):=f5(x)-x+x5

NSOLVE(f6(x)=-0.03,x,0,1)

;Approx(#38)
x=0.8815646039323056508101979831293691494913241785042641877864823864

x6:=0.8815646039323056508101979831293691494913241785042641877864823864

f7(x):=f5(x)-x6+x5

NSOLVE(f7(x)=0.03,x,0,1)

;Approx(#42)
x=0.9954541079157372145555882910132852296382355439185848107933561221

x7:=0.9954541079157372145555882910132852296382355439185848107933561221

f8(x):=f7(x)-x+x7

NSOLVE(f8(x)=-0.03,x,0,1)

;Approx(#46)
false

xv:=[x1,x2,x3,x4,x5,x6]

cvx:=[cx1,cx2,cx3,cx4,cx5,cx6]

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

;Approx(#54')
cvx1:=[0.9245140945397924410360325092686967686973157115407622699815151972,0.8~
823285334678624564137748603590629959925185094115701642966485204,0.70070164706~
48793324303255083150594475405695914171105256326979962,0.557876336534074186797~
1312759643547226586211770703361930266332392,0.3414744245679942844318381979482~
566612628185437767854097803328334,0.11022426957008181059098983092406102669343~
92148355503267920841087]

;Approx(#52')
cvx0:=[0.9110870003701001913642648099071838806767764662790637168906196531,0.8~
524024421900246711649358686586089743738385927930965417769098065,0.67457644293~
02250538934654884729767356217620528320179890680791506,0.546625407185612429030~
3172394926256906078488672949836151923835434,0.3687994079258128117588468593069~
934515840603671704480569871970894,0.18496660916510154690263175641812995747222~
43087334148945143928125]

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

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

;Approx(#58')
p0:=[24.34399822228774177079577480983131164608934379115232205277957544,31.526~
05830063111878603189004580332752894976312554737887730917741,47.57873280118548~
231572537659389564146070413750350849004527370087,56.8641915974072173955239483~
8249466883857194225431494090291189429,68.358407296222239428537116540263649662~
61216368654008575781941949,79.34081435390750857291781848164322345421917606538~
377690078341477]

;Approx(#57')
p1:=[22.40480170632382748117613871108875775910248535336423828443382226,28.075~
45851608703811085017164609224138531196350650595961259943509,45.51667561600968~
976246680603669438424044717555950046634832117572,56.0909408756567977519740761~
5082678698533231399251298288355211774,70.033270538518227238490713874851820111~
98579580230901432564714180,83.67175611589963687541312610939922452806646301350~
312569248564532]