LECTURE # 1 Complex numbers and the complex plane ... - Zahri.net

Complex Analysis, MATH 314 Summer Term 1434/2013 Dr. Mostafa Zahri

TAIBAH UNIVERSITY College of Science Department of Mathematics

LECTURE # 1  Complex numbers and the complex plane   

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Complex numbers: Definitions and Properties

A complex number can be visualized as the usual Euclidean plane: z = x + iy ∈ C is identified with the point (x; y) ∈ IR2 . • in this case 0 corresponds to the origin, • i corresponds to (0; 1). • the x and y axis of IR2 are called the real axis and imaginary axis respectively. A complex number can be written as z = = = = =

x + iy Re(z) + iIm(z) | z | eiθ | z | eiArg(z) | z | (cos(θ) + i sin(θ))

We define the absolute value of a complex number z = x + iy by | z |= (x2 + y 2)1/2 ; so that | z | is the distance from the origin to the point (x; y). In particular, the triangle inequality holds: | z + w |≤| z | + | w |; z; w ∈ C. Theorem 1. Let z1 = r1 (cos(θ1 ) + i sin(θ1 )) and z2 = r2 (cos(θ2 ) + i sin(θ2 )) Then z1 · z2 = r1 r2 (cos(θ1 + θ2 )) + i sin(θ1 + θ2 ))

Theorem 2 (De Moivres formula). z n = r n (cos(nθ)) + i sin(nθ));

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n = 1, 2, 3, 4, . . .

Complex Analysis, MATH 314 Summer Term 1434/2013 Dr. Mostafa Zahri

TAIBAH UNIVERSITY College of Science Department of Mathematics

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Problems

Problem 1. Solve in C the equation z 2 + 2 = 0. Problem 2. Solve in C the equation z 2 + 2z + 10 = 0. Problem 3. Show that the complex number z is real if and only if z = z. Problem 4. Show that the complex number z is pure imaginary if and only if z = −z. Problem 5. Show that Re(iz) = −Im(z) and Im(iz) = Re(z). Problem 6. Represent in C the numbers z1 = 1 − i and z2 = 1 + i. Problem 7. Represent in C the numbers z1 = cos(π) + i sin(π) and z2 = 2eiπ . Problem 8. Compute | z | and Arg(z) for z1 = 1 − i and z2 = 1 + 2i. Problem 9. Show that Re(z) =

z+z ; 2

and Im(z) =

Problem 10. Show that | z |2 = zz;

and

z−z . 2i

z 1 = . z | z |2

References: 1. Chrachill, Brown and Verhey R. F. Complex Variables and Applications, Me Graw-Hill, Inc, New York, 1974 2. Marden J. E. and Hoffman J.M. Basic Complex Analysis, W. H. Freeman and Company, New York, 1987. 3. Elias M. Stein & Rami Shakarchi, Complex Analysis II, Princeton University Press, 2003. 4. Elias M. Stein & Rami Shakarchi, Fourier Analysis I, Princeton University Press, 2003. 5. John M. Howie, Complex Analysis, Springer, 2007. 6. Walter Rudin, Real and Complex Analysis, 2nd ed., McGraw-Hill, 1974. 7. Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, 1979.

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