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Complex Numbers and Functions

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Complex Numbers and Functions Complex Numbers and Functions Richard Crew January 20, 2018 This is a brief review of the basic facts of complex numbers, intended for students in my section of MAP 4305/5304. I will discuss basic facts of com- plex arithmetic, limits and derivatives of complex functions, power series and functions like the complex exponential, sine and cosine which can be defined by convergent power series. This is a preliminary version and will be added to later. 1 Complex Numbers 1.1 Arithmetic. A complex number is an expression a + bi where i2 = −1. Here the real number a is the real part of the complex number and bi is the imaginary part. If z is a complex number we write <(z) and =(z) for the real and imaginary parts respectively. Two complex numbers are equal if and only if their real and imaginary parts are equal. In particular a + bi = 0 only when a = b = 0. The set of complex numbers is denoted by C. Complex numbers are added, subtracted and multiplied according to the usual rules of algebra: (a + bi) + (c + di) = (a + c) + (b + di) (1.1) (a + bi) − (c + di) = (a − c) + (b − di) (1.2) (a + bi)(c + di) = (ac − bd) + (ad + bc)i (1.3) (note how i2 = −1 has been used in the last equation). Division performed by rationalizing the denominator: a + bi (a + bi)(c − di) (ac − bd) + (bc − ad)i = = (1.4) c + di (c + di)(c − di) c2 + d2 Note that denominator only vanishes if c + di = 0, so that a complex number can be divided by any nonzero complex number. If z = a + bi, the complex conjugate or simiply conjugate of z is z¯ = a − bi: (1.5) 1 A quick calculation shows that z + w =z ¯ +w; ¯ zw =z ¯w¯ (1.6) for all z, w 2 C. From 1.3 we see that if z = a + bi then zz¯ = a2 + b2. (1.7) Note that this kind of expression appears in the denominator of 1.4, and in fact we could have written 1.4 as z zw¯ = : (1.8) w ww¯ From 1.9 we see that zz¯ ≥ 0 and zz¯ = 0 if and only if z = 0. The expression zz¯ occurs so frequently that we write p jzj = zz¯ (1.9) and then 1.8 becomes z zw¯ = : (1.10) w jwj2 One can show that jz + wj ≤ jzj + jwj; (1.11) jzwj = jzjjwj; (1.12) jzj = 0 if and only if z = 0: (1.13) In fact 1.12 can be checked by direct calculation (hint: square it first). We have already seen that 1.13 is true, and we will prove 1.11 later. The properties 1.11 to 1.13 show that j j acts like the absolute value of a real number, and accordingly we call it the absolute value or magnitude of the complex number. 1.2 Geometry. We can identify a complex number z = a + bi with a vector ha; bi in R2. The formula 1.1 then says that addition of complex numbers is the same as addition of vectors. Furthermore if we put b = 0 in 1.3 we see that multiplying a complex number by a real one is the same as scalar multiplication. To interpret 1.3 geometrically we introduce the polar form of a complex number. From ??? we see that the vector z, viewed as a line from the origin to the point z p= a + bi is the hypotenuse of a right triangle. The length of the hypotenuse is a2 + b2 = jzj and if θ is the angle the vector z makes with the x-axis, we have a = jzj cos θ; b = jzj sin θ and therefore z = jzj(cos θ + i sin θ): (1.14) This is called the polar form of z. If w = jwj(cos + i sin ) 2 is another complex number in polar form, zw = jzjjwj((cos θ cos − sin θ sin ) + i(cos θ sin + sin θ cos )) or, using the addition theorems for the cosine and sine, zw = jzjjwj(cos(θ + ) + i sin(θ + )): (1.15) In other words, to multiply two complex numbers you multiply the magnitudes and add the angles. For example the nth power of a complex number z = jzj(cos θ + i sin θ) is zn = jzjn(cos(nθ) + i sin(nθ)). From this we see that any complex number has a square root: if z = jzj(cos θ + i sin θ) then w = jzj1=n(cos(θ=n) + i sin(θ=n)) satisfies wn = z. Again, if z = jzj(cos θ + i sin θ), the angle θ is called the argument of z, written θ = arg z. A great deal of confusion can be avoided by keeping in mind the difference between an angle and a number: if θ is an angle then θ +2π is the same angle, even though it is different as a number. This causes no problems when adding, subtracting, or multiplying by an integer (it makes no sense to multiply two angles). But there is there is no unambiguous way to divide an angle by an integer. In fact if you multiply any of the angles 2π 2 · 2π (n − 1)2π θ; θ + ; θ + ; : : : ; θ + n n n by n you get nθ, and in fact these are all angles whose nth multiple is nθ. From this we see that any nonzero complex number z = jzj(cos θ + i sin θ) has n distinct nth roots, namely θ + 2kπ θ + 2kπ jzj1=n cos + i sin n n for k = 0, 1; : : : ; n − 1. In other words there is no single-valued function z1=n of z. 2 Functions and Limits 2.1 Functions. A complex function of a complex variable is a rule which to any z 2 D in some subset D ⊆ C assigns a value f(z) 2 C. The set D is the domain of the function and the set of values f(z) for all z 2 D is the range. Functions are added, substracted, multiplied and divided just like real functions. If z = x + iy then f(z) = u(x; y) + iv(x; y) for some real-valued functions u and v of x and y. From this it looks like complex calculus is going to be four times as complicated as regular calculus, but this will turn out not to be the case. 3 2.2 Limits of functions. Since the complex absolute value j j has the same properties as the usual absolute value, limits can be defined for complex functions in the same was as for real functions: we say lim f(z) = L (2.1) z!z0 if for every positive > 0 there is a δ > 0 such that if 0 < jz − z0j < δ then jf(x) − Lj < (2.2) This looks like Calculus 1 but is really more complicated since both z and f(z) have real and imaginary parts. If z = x + iy; z0 = x0 + iy0; f(z) = u(x; y) + iv(x; y); and;L = A + iB then jz − z0j ≤ jx − x0j + jy − y0j and jf(z) − Lj ≤ ju(x; y) − Aj + jv(x; y) − Bj: Using this one can show (give it a try!) that 2.1 is equivalent to two limits of two functions of two variables: lim u(x; y) = A and (x;y)!(x ;y ) 0 0 (2.3) lim v(x; y) = B: (x;y)!(x0;y0) Now in some sense it is “harder” for a function of two variables to have a limit since (x; y) can approach (x0; y0) in infinitely many different ways. Amazingly this will not turn out to be a problem for us. The rules of limits from Calculus 1 hold in this new situation: if limz!z0 f(z) = L and limz!z0 g(z) = M then lim (f(z) ± g(z) = L ± M (2.4) z!z0 lim (f(z)g(z) = LM (2.5) z!z0 f(z) L lim = if M 6= 0 (2.6) z!z0 g(z) M 2.3 Continuity. A function f(z) is continuous at z0 if 1. z0 is in the domain of f(z), 2. limz!z0 f(z) = f(z0 (in particular, the limit exists). This is the same definition as in Calculus 1. If f(z) and g(z) are continuous at z0, so are f(z) ± g(z) and f(z)g(z). So is f(z)=g(z) if g(z0) 6= 0. 4 3 Derivatives Having defined limits, we can define derivatives in the usual way: for any func- tion f(z), df f(z + h) − f(z) = f 0(z) = lim (3.1) dz h!0 h when the limit exists. If it does, we say that f(z) is differentiable at z. Like continuity this is a pointwise property, i.e. depends both on the function and the point. Since limits of complex functions are a more complicated affair then for real functions, it is in some sense harder for a complex function to have a derivative. It might therefore come as a surprise to learn that most of the functions we will deal with in fact have derivatives at every point of their domain. For example the function f(z) = z3 − z has a derivative everywhere since (z + h)3 − (z + h) − (z3 − z) f 0(z) = lim h!0 h 3z2h + 3zh2 + h3 − h = lim h!0 h = 3z2 − 1: which is even the expected result. In fact any polynomial is differentiable ev- erywhere, and the usual formula holds: n n d X X a zi = na zi−1: (3.2) dz i i i=0 i=0 You will be delighted to learn that the usual rules for computing derivatives hold when the relevant derivatives exist: (f(z) ± g(z))0 = f 0(z) ± g0(z) (3.3) (f(z)g(z)0 = f 0(z)g(z) + f(z)g0(z) (3.4) g(z)f 0(z) − f(z)g0(z) (f(z)=g(z)0 = : (3.5) g(z)2 Finally, the derivative of a constant is zero.
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