# A First Course in Complex Analysis with Applications by Dennis G. Zill By Dennis G. Zill

Written for junior-level undergraduate scholars which are majoring in math, physics, machine technology, and electric engineering.

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E e Z (18) You are encouraged to verify that the expression in (18) is the same as that given in (15). Remarks Comparison with Real Analysis We have seen in this section that if z1 is a complex root of a polynomial equation, then z2 = z¯1 is another root whenever all the coeﬃcients of the polynomial are real, but that z¯1 is not necessarily a root of the equation when at least one coeﬃcient is not real. In the latter case, we can obtain another factor of the polynomial by dividing it by z − z1 .

3 + i 3 9. −1 + i 6. 5 − 5i √ 8. −2 − 2 3i 12 10. √ 3+i In Problems 11 and 12, use a calculator to write the given complex number in polar form ﬁrst using an argument θ = Arg(z) and then using θ = Arg(z). √ √ 11. − 2 + 7i 12. −12 − 5i In Problems 13 and 14, write the complex number whose polar coordinates (r, θ) are given in the form a + ib. Use a calculator if necessary. 13. (4, −5π/3) 14. (2, 2) In Problems 15–18, write the complex number whose polar form is given in the form a + ib. Use a calculator if necessary.

C) Since y = eiθ is a solution of the diﬀerential equation, it must be obtainable from the general solution given in part (a); in other words, there must exist speciﬁc coeﬃcients c1 and c2 such that eiθ = c1 cos θ + c2 sin θ. Verify from y = eiθ that y(0) = 1 and y (0) = i. Use these two conditions to determine c1 and c2 . 29. Find a homogeneous linear second-order diﬀerential equation for which y = e−5x cos 2x is a solution. 30. (a) By diﬀerentiating Equation (13) with respect to t, show that the current in the LRC -series is described by L d2 i di 1 + R + i = E0 γ cos γt.