Using the Cauchy integral formula show that &#x222E;<!-- ∮ --> <mrow class="MJX-TeXAtom

Kallie Arroyo 2022-06-01 Answered
Using the Cauchy integral formula show that
| z | = 2 e z d z ( z 1 ) 2 ( z 3 ) = 3 2 j e π .
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Rhett Pruitt
Answered 2022-06-02 Author has 5 answers
You have
1 ( z 1 ) 2 ( z 3 ) = 1 4 1 z 1 1 2 1 ( z 1 ) 2 + 1 4 1 z 3
Now, you can use Cauchy's formula directly omitting the on | z | 2 holomorphic part belonging to 1 z 3 :
| z | = 2 e z ( z 1 ) 2 ( z 3 ) d z = 1 4 | z | = 2 e z z 1 d z 1 2 | z | = 2 e z ( z 1 ) 2 d z = 1 4 2 π i e 1 1 2 2 π i e 1 = 3 2 π i e
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-05-26
So i have this limit:
lim x x ( x 2 + 4 x 2 + 2 )
asked 2022-06-29
For f : R R and n N s. t. n 2
lim x 0 { f ( x ) + f ( n x ) } = 0
Prove/disprove :
lim x 0 f ( x ) = 0
asked 2022-04-21
How is
limh03h1h=ln3
evaluated?
asked 2022-05-25
Function with compact support whose iterated antiderivatives also have compact support
Notation: If f : R R is continuous, let us denote I f : R R its indefinite integral from 0, i.e., ( I f ) ( x ) = 0 x f ( t ) d t, and iteratively I k + 1 f = I ( I k f ).
Remark: If f is a continuous function with support contained in the open interval ]0,1[ then If has support contained in ]0,1[ iff ( I f ) ( 1 ) = 0.
Main question: Does there exist a C function f with support contained in the open interval ]0,1[ such that I k f has support contained in ]0,1[ for every k 0, or, equivalently, ( I k f ) ( 1 ) = 0 for all k 0?
Equivalent formulation: Does there exists a sequence ( f k ) k Z of C functions each with support contained in the open interval ]0,1[, such that f k 1 is the derivative of f k ?
Weaker question: Does there at least exist a continuous function f with the properties demanded in the main question?
Stronger question: Does there exist a C function f with compact support, whose Fourier transform vanishes identically on a nontrivial interval?
(A positive answer to the latter would imply a positive answer to the main question: rescale the function so its support is contained in ]0,1[, multiply it appropriately so its Fourier transform vanishes in a neighborhood of 0, and observe that the Fourier transform of I k f is, up to constants, ξ k times that of f.)
Edit: Before someone points out that the identically zero function fits the bill, I should add that I want my functions to not vanish identically.
asked 2022-06-25
lim x ( x 6 [ e 1 2 x 3 cos ( 1 x x ) ] )
Any tip on how to calculate it?
asked 2021-08-06
Graph each function and tell whether it represents exponential growth, exponential decay, or neither. y=(2.5)x
asked 2021-05-04
Use the theorems on derivatives to find the derivatives of the following function:
f(x)=3x52x45x+7+4x2

New questions