have the following polar form for a complex number z: z = jzjei arg(z): (2.2) Being an angle, the argument of a complex number is only deflned up to the addition of integer multiples of 2…. In other words, it is a multiple-valued function. This ambiguity can be resolved by deflning the principal value 4. You are given the modulus and argument of a complex number. Express the complex number in the form x + yi . (a) Modulus = 6, argument = 3 Hint: draw an Argand diagram to help. The modulus-argument form of a complex number, z , consists of the modulus, r , which is the distance to the origin, and the argument θ , which is the angle the line We can write z = x + iy as z = r cosθ + ir sinθ = r (cosθ + i sinθ), which is called the polar form of complex number. Here, r = |z| = √ (x 2 + y 2) is the modulus of z and θ is known as the argument or amplitude of z denoted as arg z. For any non-zero complex number z, there corresponds to one value of θ, in the interval [0, 2π) Definition: Complex Log Function. The function is defined as. log(z) = log( | z |) + iarg(z), where log( | z |) is the usual natural logarithm of a positive real number. Remarks. Since arg(z) has infinitely many possible values, so does log(z). log(0) is not defined. The book I'm using is Complex Variables and Applications, 9th Ed by Brown and Churchill. I'm confused about exercise 14.1.b (pg 43): For each of the functions below, describe the domain of definit The angle θ of a complex number's polar representation is its argument, z = a+ib. This is a multi-valued angle. If is the complex number z argument, then θ+2nπ, n is an integer, and will also be an argument of that complex number. The principal argument of a complex number, on the other hand, is the unique value of such that -π<θ ≤π. So the arg of z, the argument of z, is 120 degrees. And so just like that we can now think about z in polar form. So let me write it right over here. We can write that z is equal to its modulus, 2, times the cosine of 120 degrees, plus i times the sine of 120 degrees. And we could also visualize z now over here. So its modulus is 2. For example, if |z| = 2, as in the diagram, then |1/z| = 1/2. It also means the argument for 1/z is the negation of that for z. In the diagram, arg(z) is about 65° while arg(1/z) is about -65°. You can see in the diagram another point labelled with a bar over z. That is called the complex conjugate of z. May 5, 2014 at 21:47. What I meant is that arg(z) = arctan(y/x) a r g ( z) = a r c t a n ( y / x) is true if the complex number is in the first quadrant or the fourth. If the point is in the third and fourth, you need to "add" that 180. - imranfat. May 6, 2014 at 14:14. @imranfat Oh I see. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. The number a is called the real part of the complex number, and the number bi is called the imaginary part. DyN9q.