MCS 351 ENGINEERING MATHEMATICS SOLUTION OF ...

Hence we get x(t) = −a + r cos t, y(t) = −b + r sin t, −2π ≤ t ≤ 0.
18. 4(x − 1) 2 + 9(y + 2) 2 = 36 denotes an ellips.
Hence we get ellips below:
4(x − 1) 2 + 9(y + 2) 2 = 36 ⇒ ( x − 1 ) 2 + ( y + 2
3 2 )2 = 1
x 0 − 1 = 0, y 0 + 2 = 0 ⇒ x 0 = 1, y 0 = −2 ⇒ z 0 = (1, −2)
Parametric equation: x(t) = 3 cos t + 1, y(t) = 2 sin t − 2, 0 ≤ y ≤ 2π
19-29 Integration:
19. Let z = x + iy. Then f(z) = Rez = x is analytic in C. So we can use the first method:
∫ z1
z 0
f(z)dz = F (z 1 ) − F (z 0 ), F (z) analytic, F ′ (z) = f(z)
Let F (z) = x2
2 . F ′ (z) = 2 x 2 = x = f(x).
∫ 1+i
20. We can sketch C as follows:
0
f(z)dz =
∫ 1+i
0
xdz = F (1 + i) − F (0) = 1 2
If z = x + iy, then f(z) = Rez = x is analytic in C. So we can use the first method.
F (z) = x2
2 is analytic in C and F ′ (z) = 2 x 2 = x = f(x).
∫ ∫ 1+i ∫ 1+i
Rez = f(z)dz = xdz = F (1 + i) − F (0) = 1 2
C
21.
31
0
0