Question

Use the product rule to show that t-derivative of the complex-valued function f(t) = eat (cos bt + i sin bt) = e(a+bi)t is the function f(t) multiplied by a + bi. Use the previous result to find integration formulas for the real and imaginary parts of ſ f(t)dt.

Solve these two problems.

Answer 1

9. 1. Use the product rule to show that t-derivative of the complex valued function flt) = eat (coubt tisinbt) = atbilt is the function fet) multiplied by atbi. Let f(t) = eat (cos bttisinbz) f(x) = pat coubt ti at sinbt differentiate wet. ť we get f'ct)s et csinb # xb)+ cosbt ceteran til est (bxcoubt) + sinbt (qeat). } derivative) = -b et sinbt to eat coubt tibeat cosbe tigat sinbt. - at sinbt (ia-b) + at coubt (atib) - i catib) at sinbt + (atib) at coubt = (atib) [cat simbt + eat cosbt]. Catib) eat ( cosbt +isinbt). = (atib). flt) 4"+) -- Carib) f(t). hence proved. 9.2. Find integration fremulas for the real and imaginary pacts of Sfet) dt. now f(t) = eat (cosbt-tisinbt) Set) = eat Coubt tie at sinkt.

on integrating we get Stre)dt = scat cosbt ti cat sinbt)elt. S fet)dt = seat cosbtet til at sinkt dt. we tiest find seat cosbtdt. put ua castt, du= patat J eat cosbt dt = t at aroubt the seat sinbadet - ( by integration by parts.) we compute heat sinkt at Let au- estadt & u = sinbt. Seat sinkt dt = 2 at sinkt be seat coublalt ② (again by integɛation by pacts.) put ear ② in ear ③ we have. seat coobt dt = 1 at coubt + b ( 1 at isin bt-beget coubt dt) Seat coobtat de secoubt to be bt ab? Seet cosbit olt. Sectecoubteltt ? Seatcoubt de tot el cost to the indt (16) Setcositats del coste to cut zin bt seat coobtat = It be at carbtte et sin bot] I seat costata az [ 1 eet coubt that sinkt] osbt dt = - It b² lt bir 92/ 9 sinbi

we find heat sinbedt. put ear in eaf we get. Seat sinkt dt = et simb P-to the could stain stat] The bindt eerst could be seen inbt dit veterinbat at tage des erindit dit die minst - the set oubt (He ) Setsinot dt = fu simbt the encoubt :. Sects inbtelt = htbe ( te sin bl-u et coubt] Sect simblidhen be a et coubt ] GL formula for Resfet) elta is set corb tot = ahte facet coubt tb est siobt] 4 Im & fexhdt - t a pat sinbt-beat Cosby] a2462