Question

For a non-zero complex number \(z\), let \(\arg (z)\) denote the principal argument with \(-\pi<\arg (z) \leq \pi .\) Then, which of the following statement(s) is (are) FALSE? (A) \(\arg (-1-i)=\frac{\pi}{4}\), where \(i=\sqrt{-1}\) (B) The function \(f: \mathbb{R} \rightarrow(-\pi, \pi]\), defined by \(f(t)=\arg (-1+i t)\) for all \(t \in \mathbb{R}\), is continuous at all points of \(\mathbb{R}\), where \(i=\sqrt{-1}\) (C) For any two non-zero complex numbers \(z_{1}\) and \(z_{2}\), $$ \arg \left(\frac{z_{1}}{z_{2}}\right)-\arg \left(z_{1}\right)+\arg \left(z_{2}\right) $$ is an integer multiple of \(2 \pi\) (D) For any three given distinct complex numbers \(z_{1}, z_{2}\) and \(z_{3}\), the locus of the point \(Z\) satisfying the condition $$ \arg \left(\frac{\left(z-z_{1}\right)\left(z_{2}-z_{3}\right)}{\left(z-z_{3}\right)\left(z_{2}-z_{1}\right)}\right)=\pi $$ lies on a straight line

Short answer

Statement (A) is FALSE because the principal argument of \(-1-i\) is \(\frac{5\pi}{4}\), not \(\frac{\pi}{4}\). Statement (B) is FALSE due to discontinuity at \(t=0\). Statements (C) and (D) are TRUE.

Step-by-step solution

Evaluate the argument of \text{-1-i}

The complex number -1-i lies in the third quadrant of the complex plane, where both the real and imaginary parts are negative. To find the principal argument, we calculate the arctan of the imaginary part over the real part: \[\arg(-1-i) = \arctan\left(\frac{\text{imaginary part}}{\text{real part}}\right) = \arctan\left(\frac{-1}{-1}\right) = \arctan(1).\] Since it is in the third quadrant, we must add \pi to the positive result from \(\arctan(1)\) to find the principal argument, which is \[\arg(-1-i) = \pi + \arctan(1) = \pi + \frac{\pi}{4} = \frac{5\pi}{4},\] not \frac{\pi}{4}

Evaluate the continuity of the function f(t)

To assess whether the function \(f(t)=\arg(-1+it)\) is continuous, we need to see if it gives a well-defined, continuous argument for all real numbers \(t\) in the complex number \( -1+it\). This function is continuous unless the argument jumps between \( -\pi \) and \(\pi\); such jumps can occur when \(t\) changes sign, causing the complex number to cross the negative real axis, where the principal argument is defined to jump. As \(t\) ranges over all real numbers, \(f(t)\) will indeed cross the negative real axis, leading to discontinuity at \(t = 0\), thus statement (B) is FALSE.

Verify the statement about the argument of a quotient

According to the properties of arguments for non-zero complex numbers, \[\arg \left(\frac{z_1}{z_2}\right) = \arg \left(z_1\right) - \arg \left(z_2\right) + 2\pi k,\] where \(k\) is an integer that makes the result lie within \( -\pi < \arg(z) \leq \pi \). This means that, \[\arg \left(\frac{z_1}{z_2}\right) - \arg \left(z_1\right) + \arg \left(z_2\right)\] will cancel out to \(2\pi k\), which is an integer multiple of \(2\pi\). Therefore, statement (C) is TRUE.

Examine the locus of the point Z

The given condition for the locus is \[\arg \left(\frac{(z - z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)}\right) = \pi.\] This implies that the complex number inside the argument must lie on the negative real axis since only there the argument is \pi. For a complex number to lie on the negative real axis, its imaginary part must be zero, so it must be a real number. Consequently, the locus of \(Z\) satisfying the condition forms a straight line. Thus, statement (D) is TRUE.

Key concepts

Principal Argument of a Complex Number

The principal argument of a complex number is an essential concept in complex number analysis, and it helps to locate the angle a complex number makes with the positive direction of the real axis. By definition, the principal argument, denoted as \(\arg(z)\), is an angle in radians with a range of \(\pi < \arg(z) \leq \pi\). This means that for any non-zero complex number \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, the principal argument is the angle \(\theta\) that \(z\) makes with the positive real axis.

When calculating this argument, one must consider the quadrant in which the complex number is located because it determines the sign of \(\theta\). For instance, if \(z\) is in the third quadrant (both \(a\) and \(b\) are negative), the value of \(\arg(z)\) is added to \(\pi\) to adjust for the negative angle. An incorrect estimation of the quadrant leads to errors in determining the principal argument, as reflected in the solution of the exercise where \(\arg(-1-i)\) was incorrectly stated as \(\frac{\pi}{4}\) instead of the correct \(\frac{5\pi}{4}\), illustrating the importance of a proper understanding of this concept.

Remember, the principal argument remains unique to a specific complex number and is dictated by its location on the complex plane.

Continuity of Complex Functions

Understanding the continuity of complex functions is pivotal when analyzing complex-valued functions of a real variable such as \(f(t)\). A function is said to be continuous at a point if the left-hand limit, right-hand limit, and the value of the function at that point are all equal. However, the major trick in dealing with complex functions lies in the behavior of the argument function.

A function like \(f(t) = \arg(-1+it)\) may seem elusive due to the involvement of complex numbers, but it just comes down to checking whether the function provides a continuous argument across all real numbers \(t\). The principal argument can cause discontinuities, particularly where the complex number crosses the negative real axis because the argument abruptly changes from \(\pi\) to \(\-\pi\), or vice versa, which is a jump discontinuity. In the given problem, such discontinuity occurs at \(t = 0\), which is why statement (B) is false, highlighting the complexity of dealing with the continuity of functions in the realm of complex numbers.

Properties of Arguments in Complex Numbers

The behavior of arguments in complex numbers follows certain algebraic properties which are useful in simplifying complex expressions. One such property is the additive nature of arguments when dealing with the division of complex numbers, stated as \(\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2) + 2\pi k\), where \(k\) is any integer that keeps the argument within its principal range. This property enables us to predict and calculate the resultant argument when two complex numbers are divided.

In our specific problem, \(\arg(\frac{z_1}{z_2}) - \arg(z_1) + \arg(z_2)\) simplifies to \(2\pi k\), an integer multiple of \(2\pi\), thus confirming statement (C) as true. This additive property underscores how we can systematically handle arguments within equations, making once daunting tasks increasingly manageable, and reflecting the symmetrical beauty inherent in the mathematics of complex numbers.