MHB Partial Fraction Decomposition of Complex Fractions

Click For Summary
SUMMARY

The correct partial fraction decomposition of the complex fraction $$\frac{1}{z^2(1-z)}$$ is $$\frac{A}{z}+\frac{B}{z^2}+\frac{C}{1-z}$$, which includes an additional term for the linear factor in the denominator. The constants A, B, and C are assumed to be complex numbers, denoted as $$A,B,C\in\mathbb{C}$$. This decomposition allows for easier integration and manipulation of the fraction in complex analysis.

PREREQUISITES
  • Understanding of complex numbers and their properties
  • Familiarity with partial fraction decomposition techniques
  • Knowledge of algebraic manipulation of rational functions
  • Basic concepts of calculus, particularly integration
NEXT STEPS
  • Study the method of partial fraction decomposition in detail
  • Learn about complex analysis and its applications
  • Explore integration techniques involving rational functions
  • Investigate the role of complex constants in mathematical expressions
USEFUL FOR

Mathematicians, students studying calculus and complex analysis, and anyone interested in advanced algebraic techniques for simplifying rational expressions.

Dustinsfl
Messages
2,217
Reaction score
5
$$
\frac{1}{z^2(1-z)} = \frac{A}{z^2}+\frac{B}{1-z}
$$I can't figure out how to decompose this fraction.
 
Physics news on Phys.org
dwsmith said:
$$
\frac{1}{z^2(1-z)} = \frac{A}{z^2}+\frac{B}{1-z}
$$I can't figure out how to decompose this fraction.

You're missing a term. The decomposition should be of the form

\[ \frac{1}{z^2(1-z)} = \frac{A}{z}+\frac{B}{z^2}+\frac{C}{1-z} \]

Is it also to be assumed that $A,B,C\in\mathbb{C}$?
 

Thread 'How does this derivative work? And why the speed of light remains?'

Relativistic Momentum, Mass, and Energy Momentum and mass (...), the classic equations for conserving momentum and energy are not adequate for the analysis of high-speed collisions. (...) The momentum of a particle moving with velocity ##v## is given by $$p=\cfrac{mv}{\sqrt{1-(v^2/c^2)}}\qquad{R-10}$$ ENERGY In relativistic mechanics, as in classic mechanics, the net force on a particle is equal to the time rate of change of the momentum of the particle. Considering one-dimensional...
View full post »

Similar threads

Undergrad Problem with calculating projections of curl using rotation of contour
  • · Replies 9 ·
Replies
9
Views
993
MHB Integrate 3/(x(2-sqrt(x))) - No Partial Fractions
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Partial derivatives of the function f(x,y)
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad How do I format equations correctly? (Curl, etc.)
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Using Residues (Complex Analysis) to compute partial fractions
  • · Replies 4 ·
Replies
4
Views
4K
Undergrad Partial Fraction Decomposition With Quadratic Term
  • · Replies 3 ·
Replies
3
Views
2K
High School Why Can Derivatives Be Treated Like Fractions in Solving Equations?
  • · Replies 32 ·
2
Replies
32
Views
5K
MHB 206.08.05.44 partial fraction decomposition
  • · Replies 6 ·
Replies
6
Views
2K
MHB Would Partial Fractions Simplify This Integral?
  • · Replies 8 ·
Replies
8
Views
2K
MHB Evaluate the surface integral $\iint\limits_{\sum}f\cdot d\sigma$
  • · Replies 1 ·
Replies
1
Views
3K