Summary - Analysis

A complex number is of the form, where and are real numbers, and is the imaginary unit ().

• Algebraic Form: where i is the imaginary number,
• Polar Form: , where (modulus) and is the argument.

• Addition:
• Multiplication:
• Conjugation:
• Modulus:

A functionis analytic at a point if it is differentiable in some neighborhood of that point.

For to be analytic, the partial derivatives must satisfy:

The integral of a complex function along a curve:

Ifis analytic andis a closed curve within the domain of:

If is analytic inside and on a simple closed curve and is any point inside :

Functions that have a complex derivative at every point in some region.

• Polynomials:
• Exponential Function:
• Trigonometric Functions:

Integral of a complex function over a path:

Ifis analytic in a simply connected region, the integral depends only on the endpoints, not the path.

If is analytic at:

where .

Used to approximate functions near a point.

For functions analytic in an annulus:

whereare coefficients.

Laurent series are useful for dealing with functions that have singularities.

If is analytic inside and on a simple closed curve except for isolated singularities:

The residue of at is the coefficient in its Laurent series.

Generalized functions extending the concept of functions for integrating products.

The Dirac delta function.

Transforms a function of time into a function of frequency:

Signal processing, image analysis, solving partial differential equations.

Transforms a function of time into a function of a complex variable:

Solving differential equations, control theory, and systems analysis.

A variable representing outcomes of a random process.

Describes the likelihood of different outcomes.

Gives the probability that a discrete random variable is exactly equal to some value:

Binomial distribution for the number of successes intrials.

Function such that the area under the curve represents the probability of a range of outcomes:

Normal distribution representing continuous data like heights or test scores.

The average value of a random variable:

Measure of how much values differ from the mean:

Describes the probability of two random variables occurring together.

Joint distribution of height and weight.

Two variables and are independent if: