I need to re-express the term,
(z - r)^(-2)
so one over (z - r) squared, where r is a positive real number and z is a complex variable.
into multiple fractions of the form:
A(z - a)^(-1) + B(z - b)^(-1) where a and b are complex numbers and neither equals r, and A and B are functions of z.
Please help?
(z - r)^(-2)
so one over (z - r) squared, where r is a positive real number and z is a complex variable.
into multiple fractions of the form:
A(z - a)^(-1) + B(z - b)^(-1) where a and b are complex numbers and neither equals r, and A and B are functions of z.
Please help?
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Uhhh... this requires
ab = r² and a + b = 2r.
A little substitution leads to the equation
0 = b² - 2rb + r² = (b - r)² ==> b = r.
Similarly a = r. It doesn't matter if r is pure real.
Are you sure your original expression is 1/(z - r)² as opposed to 1/(z² - r²)?
Perhaps you are not aware that 1/(z - r)² is its own "partial fraction decomposition".
ab = r² and a + b = 2r.
A little substitution leads to the equation
0 = b² - 2rb + r² = (b - r)² ==> b = r.
Similarly a = r. It doesn't matter if r is pure real.
Are you sure your original expression is 1/(z - r)² as opposed to 1/(z² - r²)?
Perhaps you are not aware that 1/(z - r)² is its own "partial fraction decomposition".