HSC 2015 MX2 Integration Marathon (archive) (1 Viewer)

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braintic

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Re: MX2 2015 Integration Marathon

Also, isn't the principle square root of a complex number a+bi (a, b real, a not negative if b = 0) defined as the one with positive real part (this definition fails if the complex number is a negative real number).
Defined by whom?
 

InteGrand

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Re: MX2 2015 Integration Marathon

Defined by whom?
Defined in mathematics (not sure which person defined it like this first), like how square root of a positive number is defined as the positive value, e.g. principal value of is defined to be 2, not -2.

Wikipedia explains it here, http://en.wikipedia.org/wiki/Square_root#Principal_square_root_of_a_complex_number, and I've seen it on other sites, but haven't checked any complex analysis textbooks yet, I'll try and find some maybe.

There is a geometric interpretation on the second answer here: http://math.stackexchange.com/questions/44406/how-do-i-get-the-square-root-of-a-complex-number
 
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braintic

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Re: MX2 2015 Integration Marathon

Defined in mathematics (not sure which person defined it like this first), like how square root of a positive number is defined as the positive value, e.g. principal value of is defined to be 2, not -2.

Wikipedia explains it here, http://en.wikipedia.org/wiki/Square_root#Principal_square_root_of_a_complex_number, and I've seen it on other sites, but haven't checked any complex analysis textbooks yet, I'll try and find some maybe.

There is a geometric interpretation on the second answer here: http://math.stackexchange.com/questions/44406/how-do-i-get-the-square-root-of-a-complex-number
Just wondering what is the point of defining a principal square root in the first place? Why should one take precedence over the other, especially when the choice seems completely arbitrary?
 

InteGrand

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Re: MX2 2015 Integration Marathon

Just wondering what is the point of defining a principal square root in the first place?
It's probably done so that there is a meaning to what is referred to by the expression that mathematicians can agree on, which is probably useful in complex analysis, like how it's useful with real numbers. If we didn't have principal square roots for positive real numbers, we'd have to keep saying which one we're referring to if we were using one in an equation or something, whereas with a principal root defined, we don't need to do this. This may also be useful for complex numbers for those doing complex analysis.


Why should one take precedence over the other, especially when the choice seems completely arbitrary?
Well, for square roots of positive numbers, it's more natural to define the positive square root as the principal value. Otherwise, we'd have referring to -2 and refer to +2. While there wouldn't be anything mathematically wrong with this (it's just notation), it's more natural and less confusing to have the notation that has a negative sign to be the negative number, so we define the principal value of the square root of a positive number as the positive square root.

Defining the principal value of a complex number as the one with positive real part is maybe done because then the definition of the principal value of the square root of a positive real number is consistent with this, so the complex number definition works for all complex numbers except for the negative real numbers (since these have square roots with 0 real part). For a negative number , maybe the principal square root is taken to be (where refers to the principal value, which is the positive square root), since is defined as the principal value of .
 

seanieg89

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Re: MX2 2015 Integration Marathon

Yeah for negative reals, the principal square root is just an imaginary number with positive imaginary part.

More generally, the principal argument of a complex number z is the argument in the interval (-pi,pi]. (Note that any complex number will have at least one value of its arg in this range.)

This gives us the principal logarithm



Generally, we have





as the means of defining complex powers as a multi-function. But if we replace arg by principal arg, log becomes single-valued and we get a notion of arbitrary principal powers. Certain familiar properties of positive real exponentiation break though, but it is still convenient to have an single-valued operator sometimes. (Say we only wanted to prove the existence of something, and our work required the extraction of a root for example.)
 

InteGrand

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Re: MX2 2015 Integration Marathon

Firstly, we note that (we will use this later; this integral can be found, for example, by using integration by parts: integrate the 1 and differentiate the ).

Let .

Then .


For , , so for , we have


.

Let . Then and , so we get

.

Then













, for some constant c.

For , we would get the negative of this, since at the start, we would have had .

So overall, we have, for :

 
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Ekman

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Re: MX2 2015 Integration Marathon

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