Office hours: Tuesdays 5-6 PM (undergrad class preference), Wednesdays 1-2 PM (free-for-all), Fridays 12-1 PM (this class preference)

For September 3-4, please let me know in advance if you want to make use of office hours; I have a visitor…

Related research talk on network analysis on Tuesday, September 3 at 4 PM in Amos Eaton 214•Opportunities to interact with speaker Mason Porter (Oxford); let me know if you're interested.•

Special student colloquium talk on mathematics of social networks on Wednesday, September 4 at 2 PM in Amos Eaton 214.

Dettman 1.1-1.3•Ahlfors 1.1-1.5•Ablowitz 1.1, 2.2•

Readings:

(for example define ○

One can show that, "up to isomorphism," the way we defined complex numbers is the only way to define a field structure on a two-dimensional real vector space.

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In particular, this must also be (up to isomorphism) the way to extend the real numbers to a field for which the equation x2+1=0 is solvable. And in fact, they turn out to be identical, not just up to isomorphism.

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Last time we showed that the complex numbers, together with their familiar arithmetic, can be constructed algebraically simply as a field which is also a two-dimensional vector space over the real numbers. There was no need to discuss notions of square of negative numbers to define the complex numbers in this way.

General algebraic field theory gives you all the usual rules about arithmetic with complex variables, these work more or less the same way as for real variables.

But the rules will become more subtle once we do algebra, not just arithmetic.

To prepare, it's useful to invoke the polar coordinate representation of complex numbers.

Complex arithmeticMonday, August 26, 20135:18 PM

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The conversion formulas:

Modulus

Argument (amplitude?!)

It is essential in complex variable theory to treat multi-valued functions as they are, and if one is making specific single-valued functions out of multi-valued functions, to distinguish these.

So, in particular, we define a single-valued version of this argument by definingArg z as the principal value of arg z as follows: Arg z is the unique value from the list arg z such that

Also, arg z =

The reverse relationships is more straightforward:

X = r cos

For example,

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Notice that we can define circles in the complex plane by:

Well why introduce polar coordinates if it creates the multivalued pain?

Among other things, multiplication of complex numbers is easier in polar than Cartesian coordinates.

Addition is easier in Cartesian coordinates.

Multiplication:

Moduli multiply and arguments add:

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As multivalued functions! Not true for single-valued version.

In particular, if we take integer powers of complex numbers:

The operations we have defined so far have geometric interpretations as transformations on the complex plane.

Addition by a complex number:

Gives translation by z

Multiplication by a complex number:

Produces a rotary stretching.Stretches by |z|Rotates by arg z

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Inversion:

Amounts to a reflection across the real axisAnd a rescaling by the inverse modulus.

Power laws:

Moduli are raised to the corresponding power.Arguments are stretched by the corresponding power.Note the image could intersect itself.

Complex conjugation:

Reflection across the real axis.

The next step of development is to be able to discuss algebraic roots of complex numbers, i.e., to invert the power law map. But we know the power law map is not injective, so roots will not be uniquely defined.

is to be defined as the inverse function of

But will need to be interpreted as a multi-valued function.

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function.

Moduli of equal complex numbers must be equal:

The arguments can either be equated as multi-valued functions or allowed to differ by multiples of 2π if interpreted as single-valued variables.

Without the multi-valued interpretation, we'd have to say:

Let's write this in terms of moduli and the multi-valued and single-valued argument functions.

Is a subset of n consecutive integers containing 0.

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Example:

This winds up given 4 distinct complex numbers (i.e., the same as the order of the root). To write these down, one simply chooses 4 values of the argument that give distinct values:

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