Definition of Principal Value

In summary: Apparently the textbook is using a different convention.In summary, the textbook defines the principal value of z^c, where both variables are complex, to be e^{c\; \text{Log }z}, where \text{Log} is the principal value branch of the complex logarithm. However, when calculating z^c with z = i and c = 3, it may not always result in the principal value. This is because the textbook reduces the value without explanation. In general, any complex number to a positive integer power is single valued. However, if z^c has both variables complex, such as i^{3i}, then the principal value is e^{-3\pi}, according to the textbook's
  • #1
hbweb500
41
1
I am studying complex variables with Brown and Churchill. In it, they define the principal value of [tex]z^c[/tex], with both variables complex, to be [tex] e^{c\; \text{Log }z} [/tex], where [tex]\text{Log}[/tex] is the principle value branch of the complex logarithm.

Now, suppose [tex] z = i [/tex] and [tex] c = 3 [/tex]. We know that [tex] \text{Arg } i = \frac{\pi}{2} [/tex], so [tex] z^c = i^3 = e^{3 \pi / 2} [/tex]. But is this really the principal value? Why don't we say [tex] e^{- \pi/2} [/tex] is the principal value?

I ask because it seems like that is what the textbook does in one of its examples: it calculates
[tex]z^c[/tex] to be something with an angle outside of [tex]-\pi < \theta \leq \pi [/tex], and just reduces it without explanation.

So, when finding the principle value of [tex]z^c[/tex] after we have done the calculation, or is simply using the principle value branch logarithm enough?
 
Mathematics news on Phys.org
  • #2
Log already chooses a principal value by only taking i Arg z for the complex part. Hence it isn't necessary to introduce further conventions to get an principal value for z^c.

Anyhow, your exponential should have an i upstairs I think, so that it doesn't matter whether the exponent is -i pi/2 or i 3 pi/2.

Now if you were to take Arg of the exponential then I suppose you'd have to return a value in (-pi, pi].
 
Last edited:
  • #3
Yes, you are missing an "i" in the numerator [itex]i^3= e^{3i\pi/2}= -i[/itex] as you get by straight forward multiplication: [itex]i^3= (i^2)i= -i[/itex].

Doing it as [itex]e^{3 log(i)}[itex], [itex]log(i)= i\pi/2+ 2k\pi i[/itex] so that [itex]i^3= e^{3 log(i)}= e^{3i\pi/2+ 6ki\pi}= e^{3i\pi/2}e^{6ki\pi}[/itex]. But e to any even multiple of [itex]i\pi[/itex] is 1 so that all "branches" give the same thing. More generally, any complex number to a positive integer power is single valued.

But, since you said "[itex]z^c[/itex] with both variables complex", did you mean [itex]i^{3i}[/itex]. In that case, [itex]i^{3i}= e^{3i log(i)}[/itex] and now [itex]i= e^{i\pi/2}[/itex] so that [itex]log(i)= i\pi/2+ 2k\pi[/itex] as before, [itex]3i \log(i)= -3\pi/2+ 5k\pi[/itex] and, finally, [itex]i^{3i}= e^{-3\pi}e^{5k\pi}[/itex].

Taking k= 0 gives [itex]i^{3i}= e^{-3\pi}[/itex] which is the smallest positive value. Normally, the "principal value" of a calculation is the non-real value with smallest argument. When all values are real, it is the smallest positive value.
 
Last edited by a moderator:

1. What is the definition of principal value?

The principal value, also known as the principal branch, is the value of a complex function that is chosen as the "main" value when multiple values are possible. It is often denoted as PV or Arg.

2. How is principal value different from other values of a complex function?

Unlike other values of a complex function, the principal value is chosen to be continuous and smooth. This means that it avoids discontinuities and jumps that may occur with other values.

3. What is an example of a function where principal value is used?

The most common example of a function where principal value is used is the complex logarithm function, Log(z). This function has infinitely many values, but the principal value is chosen to be the value with an argument between and π.

4. How is principal value useful in mathematics and science?

Principal value is useful in mathematics and science because it provides a standardized, consistent value for a complex function in cases where multiple values are possible. This allows for easier calculations and analysis of functions.

5. Can principal value be applied to real-valued functions?

No, principal value is only applicable to complex-valued functions. Real-valued functions do not have multiple values, so there is no need for a principal value.

Similar threads

  • General Math
Replies
5
Views
636
Replies
7
Views
1K
Replies
13
Views
3K
Replies
4
Views
685
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
Replies
1
Views
683
Replies
4
Views
788
  • General Math
Replies
1
Views
750
Replies
3
Views
908
Replies
9
Views
1K
Back
Top