Power Series Representations to Know for Intro to Complex Analysis

Power series representations are key in complex analysis, allowing us to express functions as infinite sums. This approach includes Taylor and Maclaurin series, which approximate functions, and extends to geometric, exponential, and trigonometric series, revealing their convergence properties.

1. Taylor series

   • Represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point.
   • The general form is ( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n ).
   • Provides a powerful tool for approximating functions near the point ( a ).

2. Maclaurin series

   • A special case of the Taylor series centered at ( a = 0 ).
   • The general form is ( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n ).
   • Useful for simplifying calculations for functions around zero.

3. Geometric series

   • A series of the form ( \sum_{n=0}^{\infty} ar^n ) converges to ( \frac{a}{1 - r} ) for ( |r| < 1 ).
   • Fundamental in deriving other series and understanding convergence.
   • Forms the basis for many power series representations.

4. Exponential function series

   • The series for ( e^x ) is given by ( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} ).
   • Converges for all ( x ), making it an entire function.
   • Illustrates the concept of power series representing transcendental functions.

5. Sine and cosine series

   • The series for sine is ( \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} ).
   • The series for cosine is ( \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} ).
   • Both series converge for all ( x ) and are essential in Fourier analysis.

6. Logarithmic series

   • The series for ( \ln(1+x) ) is ( \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n} ) for ( |x| < 1 ).
   • Useful for approximating logarithmic functions and understanding their behavior near 0.
   • Highlights the importance of convergence in power series.

7. Binomial series

   • The series for ( (1+x)^k ) is ( (1+x)^k = \sum_{n=0}^{\infty} \binom{k}{n} x^n ) for ( |x| < 1 ).
   • Generalizes the binomial theorem to non-integer exponents.
   • Important for combinatorial applications and series expansions.

8. Radius of convergence

   • Determines the interval within which a power series converges.
   • Can be found using the ratio test or root test.
   • Essential for understanding the behavior of series and their limits.

9. Abel's theorem

   • States that if a power series converges at a point on its boundary, it converges uniformly on compact subsets within that boundary.
   • Important for establishing the continuity of functions defined by power series.
   • Provides insight into the behavior of series at the edge of their convergence.

10. Laurent series

    • Extends the concept of power series to include terms with negative powers, useful for functions with singularities.
    • The general form is ( f(z) = \sum_{n=-\infty}^{\infty} a_n (z - a)^n ).
    • Essential for complex analysis, particularly in residue theory and contour integration.