Using the definition of complex derivative, evaluate f(z) expression using derivative operation based on limiting case as lim_{triangle z ->0}

arenceabigns 2021-09-15 Answered

3) Answer the following questions considering the complex functions given below.
a) Using the definition of complex derivative, evaluate f(z) expression using derivative operation based on limiting case as \(\displaystyle\lim_{{\triangle{z}\rightarrow{0}}}\)
a.1 \(\displaystyle{f{{\left({z}\right)}}}={\frac{{{1}}}{{{z}^{{2}}}}},{\left({z}\ne{}{0}\right)}\)

Expert Community at Your Service

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

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

wheezym
Answered 2021-09-16 Author has 13417 answers

Step 1
a.1 We have the definition complex derivative -
\(\displaystyle{f}`{\left({z}\right)}=\lim_{{\triangle{z}\rightarrow{0}}}{\frac{{{f{{\left({z}+\triangle{z}\right)}}}-{f{{\left({z}\right)}}}}}{{\triangle{z}}}}\)
Given, \(\displaystyle{f{{\left({z}\right)}}}={\frac{{{1}}}{{{z}^{{2}}}}},{\left({z}\ne{}{0}\right)}\)
Then,
\(\displaystyle{f}`{\left({z}\right)}=\lim_{{\triangle{z}\rightarrow{0}}}{\frac{{{\frac{{{1}}}{{{\left({z}+\triangle{z}\right)}^{{2}}}}}-{\frac{{{1}}}{{{z}^{{2}}}}}}}{{\triangle{z}}}}\)
\(\displaystyle=\lim_{{\triangle{z}\rightarrow{0}}}{\frac{{{z}^{{2}}-{\left({z}+\triangle{z}\right)}^{{2}}}}{{{\left({z}+\triangle{z}\right)}^{{2}}{z}^{{2}}\triangle{z}}}}\)
\(\displaystyle=\lim_{{\triangle{z}\rightarrow{0}}}{\frac{{{1}}}{{{z}^{{2}}}}}{\frac{{{0}-{2}{\left({z}+\triangle{z}\right)}{1}}}{{{\left({z}+\triangle{z}\right)}^{{21}}+\triangle{z}{z}{\left({z}+\triangle{z}\right)}{1}}}}\)
\(\displaystyle={\frac{{{1}}}{{{z}^{{2}}}}}\times{\frac{{-{2}{z}}}{{{z}^{{2}}+{0}}}}\)
\(\displaystyle={\frac{{{1}}}{{{z}^{{2}}}}}\times{\frac{{-{2}}}{{{z}}}}\)
\(\displaystyle=-{2}{z}^{{-{3}}}\)
\(\displaystyle\text{Hence, }\ {\left({\frac{{{1}}}{{{z}^{{2}}}}}\right)}^{{1}}=-{2}{z}^{{-{3}}},{z}\ne{}{0}\)
Step 2
Given, \(\displaystyle{f{{\left({z}\right)}}}={\frac{{{z}}}{{{z}+{1}}}},{\left({z}\ne{}-{1}\right)}\)
\(\displaystyle\text{Then, }\ {f}`{\left({z}\right)}=\lim_{{\triangle{z}\rightarrow{0}}}{\frac{{{\frac{{{z}+\triangle{z}}}{{{z}+\triangle{z}+{1}}}}-{\frac{{{z}}}{{{z}+{1}}}}}}{{\triangle{z}}}}\)
\(\displaystyle=\lim_{{\triangle{z}\rightarrow{0}}}{\frac{{{\left({z}+\triangle{z}\right)}{\left({z}+{1}\right)}-{z}{\left({z}+\triangle{z}+{1}\right)}}}{{{\left({z}+\triangle{z}+{1}\right)}{\left({z}+{1}\right)}\triangle{z}}}}\ \ {\left({\frac{{{0}}}{{{0}}}}\ \text{ form}\right)}\)
\(\displaystyle=\lim_{{\triangle{z}\rightarrow{0}}}{\frac{{{1}}}{{{z}+{1}}}}{\frac{{{\left({z}+{1}\right)}\cdot{1}-{z}\cdot{1}}}{{{\left({z}+\triangle{z}+{1}\right)}\cdot{1}+\triangle{z}\cdot{1}}}}\)
\(\displaystyle=\lim_{{\triangle{z}\rightarrow{0}}}{\frac{{{1}}}{{{z}+{1}}}}{\frac{{{z}+{1}-{z}}}{{{\left({z}+\triangle{z}+{1}\right)}+\triangle{z}}}}\)
\(\displaystyle={\frac{{{1}}}{{{\left({z}+{1}\right)}^{{2}}}}}\)
\(\displaystyle\text{Hence, }\ {\left({\frac{{{z}}}{{{z}+{1}}}}\right)}^{{1}}={\frac{{{1}}}{{{\left({z}+{1}\right)}^{{2}}}}},{z}\ne{}-{1}\)

Have a similar question?
Ask An Expert
34
 

Expert Community at Your Service

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

Relevant Questions

asked 2021-09-11
Write down the definition of the complex conjugate, \(\displaystyle\overline{{{z}}}\ \text{ ,if }\ {z}={x}+{i}{y}\) where z,y are real numbers. Hence prove that, for any complex numbers w and z,
\(\displaystyle\overline{{{w}\overline{{{z}}}}}=\overline{{{w}}}{z}\)
asked 2021-09-14

1. Given a complex valued function can be written as \(f(z) = w = u(x,y) + iv(x,y)\), where w is the real part of w and v is the imaginary part of v. Using algebraic manipulation figure out what wu and v is if
\(\displaystyle{f{{\left({z}\right)}}}=\overline{{{z}}}{\left({1}-{e}^{{{i}{\left({z}+\overline{{{z}}}\right)}}}\right)}\)
Here \(\displaystyle\overline{{{z}}}\) denotes the complex conjugate of z.
2. Using the concept of limits figure out what the second derivative of \(\displaystyle{f{{\left({z}\right)}}}={z}{\left({1}-{z}\right)}\) is.
3. Use the theorems of Limits that have been discussed before to show that
\(\displaystyle\lim_{{{z}\rightarrow{1}-{i}}}{\left[{x}+{i}{\left({2}{x}+{y}\right)}\right]}={1}+{i}.{\left[\text{Hint: Use }\ {z}={x}+{i}{y}\right]}\)

asked 2021-09-07
Determine an elliptic cylinder such that the length of its major axis length is 5/3 times of its minor axis. Find the complex potential , F(z) , and complex velocity, W(z) , for a uniform flow stream (at zero angle of attack) past the cylinder without circulation. You can use a=1 , and U=20 for your baseline flow parameters.
asked 2021-09-17
There are four washing machines in an apartment complex: A, B, C, D. On any given day the probability that these machines break down is as follows:
P(A) = 0.04, P(B) = 0.01, P(C) = 0.06, P(D) = 0.01 .
Assume that the functionality of each machine is independent of that of others. What is the probability that on a given day at least one machine will be working?
asked 2021-09-04
Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{4}}-{5}{x}^{{3}}-{36}{x}^{{2}}+{272}{x}-{448}\)
All complex zeros are=?
asked 2021-09-04

(a)Show that for all complex numbers z and w
\(\displaystyle{\left|{z}-{w}\right|}^{{2}}+{\left|{z}+{w}\right|}^{{2}}={2}{\left|{z}\right|}^{{2}}+{2}{\left|{w}\right|}^{{2}}\)
(b) Let u,v be complex numbers such that \(\displaystyle{\left|{u}\right|}={\left|{v}\right|}={1}{\quad\text{and}\quad}{\left|{u}-{v}\right|}={2}\) Use part (a) to express u in terms of v.

asked 2021-09-08
Is the complex number \(\displaystyle{z}={e}^{{2}}{e}^{{{1}+{i}\pi}}\) pure imaginary? Is it real pure?
Write its imaginary part, its real part, its module and argument. Write its complex conjugate.
Calculate and write the result in binomial form.

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question
...