For z, z^' ≠0, log(z z^2) and logz+logz^* differ by a multiple of (a) n π(b) 2 n π(c) 0 (d) (2 n

step 1 So they're continuing this caution here from previous two videos. You can refer previous two vid

For z, z^' ≠0, log(z z^2) and logz+logz^* differ by a...

Key Concepts

Complex Conjugation and Logarithms

The complex conjugate interacts with the logarithm in a way that the logarithm of a conjugate relates to the logarithm of the original number by a change in the sign of the imaginary part. This relation is important to understand when comparing sums of logarithms (involving both a number and its conjugate) to the logarithm of a product, as any difference may again reflect the issue of the multivalued nature of the complex logarithm.

Complex Logarithm

The complex logarithm extends the notion of the logarithm to the complex plane. Unlike its real counterpart, it is a multivalued function because the complex argument (angle) is only defined modulo 2?. Thus, when taking the logarithm of a nonzero complex number, one must account for the fact that there are infinitely many values differing by integer multiples of 2?i.

Branches and Branch Cuts

When dealing with the complex logarithm, one typically selects a single-valued branch by imposing a restriction on the argument (typically to lie within an interval of length 2?, like (-?, ?]). This is referred to as choosing a branch or a branch cut. However, when performing algebraic manipulations, the choice of branch must be handled carefully because different choices can lead to differences by multiples of 2?i.

Addition Property for Logarithms

In ordinary arithmetic, the logarithm of a product is the sum of the logarithms, i.e., log(ab) = log a + log b. However, in the complex case this formula holds only up to the addition of an integer multiple of 2?i due to the multivalued nature of the logarithm. Thus, when adding complex logarithms, one must take into account these possible extra multiples of 2?i, which are essential in understanding any discrepancy between expressions like log(ab) and log a + log b.

Transcript

00:01 So they're continuing this caution here from previous two videos.

00:06 You can refer previous two videos.

00:08 We're going to hear this information.

00:10 And also we are given here z and z -d -dash.

00:13 To complex numbers, that's not equal to 0.

00:18 We're going to find out here log of z -z -d -dash and log z plus log z -ddddddd differ by a multiple of what.

00:28 So i'm going to check that.

00:30 So for caution like this, simplify that.

00:33 If you just find out here, log z z dash, and then log z plus log z dash.

00:39 So let's first find out that.

00:44 So that could be lengthy.

00:46 So what we can do, we just simplify it.

00:48 We can just put some values of z and z dash, and then we can check it.

00:52 I'm going to the given relation.

00:55 So we can just go like this.

01:00 We have here it is z and z dash.

01:03 So we can just take it like this.

01:04 We take z as negative 1 and z dash as negative 2 so we have z as negative 1 and z dash that equals negative 2 so let's say we have and now we just find out here low z z dash so that's coming out to be low z z dash that equals for the value z is negative 1 3 dash is negative 2 so is coming out to be log of mode, then z z dash that's coming out to be two.

01:52 Maybe have more two plus iota, argument z z dash.

02:01 So argument of two.

02:04 Next, we're to check for log z plus log z dash.

02:11 So log z plus log z dash, if we do that...

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