Functions of a Complex Variable - Tutorial Text No. 2

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For example, John Wallis wrote, "These Imaginary Quantities as they are commonly called arising from the Supposed Root of a Negative Square when they happen are reputed to imply that the Case proposed is Impossible" Wells , p. Here, is known as the complex modulus or sometimes the complex norm and is known as the complex argument or phase.

The plot above shows what is known as an Argand diagram of the point , where the dashed circle represents the complex modulus of and the angle represents its complex argument. Historically, the geometric representation of a complex number as simply a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, "imaginary" numbers became accepted partly through their visualization.

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Unlike real numbers, complex numbers do not have a natural ordering, so there is no analog of complex-valued inequalities. This property is not so surprising however when they are viewed as being elements in the complex plane , since points in a plane also lack a natural ordering.

Möbius transformations

The absolute square of is defined by , with the complex conjugate , and the argument may be computed from. The real and imaginary parts are given by. A power of complex number to a positive integer exponent can be written in closed form as. Complex numbers may also be taken to complex powers. For example, complex exponentiation obeys. Abramowitz, M. New York: Dover, pp.

Module Details

Arfken, G. Mathematical Methods for Physicists, 3rd ed.

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Orlando, FL: Academic Press, pp. Bold, B. Courant, R. Oxford, England: Oxford University Press, pp. Ebbinghaus, H. New York: Springer-Verlag, Krantz, S. Mazur, B. Farrar, Straus and Giroux, Morse, P. New York: McGraw-Hill, pp. The world belongs, however, to those who can look at the glass and say: What's up with this glass?

Excuse me? This is my glass?

Data Types and Variables

I don't think so. My glass was full! And it was a bigger glass! We also like to thank Denise Mitchinson for providing the style sheet of this website. We want to keep it like this. You can help with your donation: The need for donations Job Application bodenseo is looking for a new trainer and software developper. You need to live in Germany and know German. Find out more! If you want to acquire special knowledge in Text Processing and Text Classification, then "Python Text Processing Course" will be the right one for you. You can book Bernd Klein for on-site Python courses as well.

Variables and Data Types A remarkable aspect of Python is that not only the value of variable can change during program execution but the type as well.

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This means that there can't be no type declaration for variables. Variable Declaration There is no variable declaration in Python! You just assign a value to a variable and it comes into existence. But never use a variable name without having assigned a value beforehand. Complex Numbers Complex numbers are part of the core language of Python. As the name implies, a variable is something which can change.

A variable is a way of referring to a memory location used by a computer program. A variable is a symbolic name for this physical location. This memory location contains values, like numbers, text or more complicated types. A variable can be seen as a container or some say a pigeonhole to store certain values. While the program is running, variables are accessed and sometimes changed, i. In strongly-typed languages every variable must have a unique data type.

In Java or C, every variable has to be declared before it can be used.

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  5. Declaring a variable means binding it to a data type. Declaration of variables is not required in Python.

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    If there is need of a variable, you think of a name and start using it as a variable. Another remarkable aspect of Python: Not only the value of a variable may change during program execution but the type as well. You can assign an integer value to a variable, use it as an integer for a while and then assign a string to the variable. It should be "read" or interpreted as "is set to", meaning in our example "the variable i is set to 42".

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    7. Identifiers Variables and identifiers are very often mistaken as synonyms. In simple terms: The name of a variable is an identifier, but a variable is "more than a name". A variable has a name, in most cases a type, a scope, and above all a value.

      Besides this, an identifier is not only used for variables. An identifier can denote various entities like variables, types, labels, subroutines or functions, packages and so on. Naming Identifiers of Variables Every language has rules for naming identifiers.