By Yue Kuen Kwok

This introductory textual content on complicated variable tools has been up to date with much more examples and exercises.

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**Example text**

In fact, the function may be expressed as a quadratic polynomial in z, where z2 − zw + 1 = 0. For a given value of w, there are in general two values of z which satisfy the above relation. Therefore, the function w(z) = z + 1z is many-to-one. (b) For any two complex numbers z1 and z2 , we have w(z1 ) = w(z2 ) iz1 + 4 iz2 + 4 ⇐⇒ = 2z1 + 3i 2z2 + 3i ⇐⇒ 2iz1 z2 + 8z2 − 3z1 + 12i = 2iz1 z2 + 8z1 − 3z2 + 12i ⇐⇒ z1 = z2 . iz+4 is one-to-one. Therefore, the function w(z) = 2z+3i (c) Consider the following relation: z = reiθ = rei(2π+θ) .

Prove the commutative, associative and distributive rules for the addition and multiplication of complex numbers. 5. Show that if the product of two complex numbers is zero, then at least one of them is zero. 6. Show that (1 + cos θ + i sin θ)n = 2n cosn θ nθ nθ cos + i sin 2 2 2 . 7. Use de Moivre’s theorem to show that sin(2m + 1)θ = sin2m+1 θ Pm (cot2 θ ), for 0 < θ < π , 2 where m Pm (x) = 2m+1 m−k (−1)k C2k+1 x , k=0 Crn = n! r! Hence, deduce that m cot2 k=1 kπ m(2m − 1) = . 8. A complex number, represented by x + iy, may also be visualized as a 2 × 2 matrix x −y y x .

0 ∞ Topologically, any set of the form {z : |z| > R}, where R ≥ 0, is called a neighborhood of ∞. To approach the point at infinity, we let |z| increase without bound while Arg z can assume any value. Note that the open upper half-plane Im z > 0 does not contain the point at infinity since Arg z is restricted to take value in the interval (0, π ). The modulus value of the point of infinity is infinite while the argument value is indeterminate. 9). We call the point of contact the south pole (denoted by S) and the point diametrically opposite S the north pole (denoted by N ).