a) Find the Maclaurin series expansion for: ) (ર): 4ૉ+) 32452-2 (2) f(z) = 1- Cosz Z² What is the radius of convergence? (5) Find the Taylor series expansion about Z₁ = πi for flz)=e² What is the radius of convergence? Ex 15.3: Find the power series expansion of the function. about the point to = 0 and determine the radices. $(z) = 2z-3 of convergence. Sol: Observe that I can be written in the form of a convergent geometric series as follows: -1. £ £ ( 2² )" = रेस्-डे n=o The series expansion is valid only if | 22 | < 1 ⇒ the radius of convergence of this series. R = 3/₁ is Some natural questions to pose at this point are: What happens if we asked to find the power series representation of a function of that cannot be expressed as a geometric series? Is there a general method by which we can find the power series representation of an arbitrary function? How do we even know that a function admits a power Series representation ? Now we will partially answer these questions. Thm 13.1 (Taylor series). Let f(z) be an analytic function on a domain √ let Z = aεr be some point in J. Assume that a Circle C of radius r centered at Z=a is contained in J. Then, there exists a unique power series with radius of convergence at least r that converges to f(z): f(z) = = \(^) (a) (z-a)" | f(n)(a)| ≤ f(n) M n=o where n! Mis the maximum of (f(z) on the circle 1Z-al=r= 14(z)| ella z M = max 1z-al=r n=2 > The series on the right is called the Taylor series for I at a. Rmk For Q=0 the series is called the Maclaurin series. Namely, of is differentiable arbitrarily many times at a and this is a unique representation of f as power series on C. Note: We conclude that if f is once differentiable, then can be written as power series, then we can dedece: By Prop 12.2: f'(z) = { f(n)(a) (z-a)n-1 n=1 (n-1)! p(n) (a) (n-2)! n n и 2 и 1/2 (²) "z" = -2 ²3 tl Z² n=o (z-a)n-2 2 and so on for all ZEC.
a) Find the Maclaurin series expansion for: ) (ર): 4ૉ+) 32452-2 (2) f(z) = 1- Cosz Z² What is the radius of convergence? (5) Find the Taylor series expansion about Z₁ = πi for flz)=e² What is the radius of convergence? Ex 15.3: Find the power series expansion of the function. about the point to = 0 and determine the radices. $(z) = 2z-3 of convergence. Sol: Observe that I can be written in the form of a convergent geometric series as follows: -1. £ £ ( 2² )" = रेस्-डे n=o The series expansion is valid only if | 22 | < 1 ⇒ the radius of convergence of this series. R = 3/₁ is Some natural questions to pose at this point are: What happens if we asked to find the power series representation of a function of that cannot be expressed as a geometric series? Is there a general method by which we can find the power series representation of an arbitrary function? How do we even know that a function admits a power Series representation ? Now we will partially answer these questions. Thm 13.1 (Taylor series). Let f(z) be an analytic function on a domain √ let Z = aεr be some point in J. Assume that a Circle C of radius r centered at Z=a is contained in J. Then, there exists a unique power series with radius of convergence at least r that converges to f(z): f(z) = = \(^) (a) (z-a)" | f(n)(a)| ≤ f(n) M n=o where n! Mis the maximum of (f(z) on the circle 1Z-al=r= 14(z)| ella z M = max 1z-al=r n=2 > The series on the right is called the Taylor series for I at a. Rmk For Q=0 the series is called the Maclaurin series. Namely, of is differentiable arbitrarily many times at a and this is a unique representation of f as power series on C. Note: We conclude that if f is once differentiable, then can be written as power series, then we can dedece: By Prop 12.2: f'(z) = { f(n)(a) (z-a)n-1 n=1 (n-1)! p(n) (a) (n-2)! n n и 2 и 1/2 (²) "z" = -2 ²3 tl Z² n=o (z-a)n-2 2 and so on for all ZEC.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
By looking at the notes for example, I need help with the circled questions.
Transcribed Image Text:a) Find the Maclaurin series expansion for:
) (ર): 4ૉ+)
32452-2
(2) f(z) = 1- Cosz
Z²
What is the radius of convergence?
(5) Find the Taylor series expansion about Z₁ = πi for flz)=e²
What is the radius of convergence?
Transcribed Image Text:Ex 15.3: Find the power series expansion of the function.
about the point to = 0 and determine the radices.
$(z) =
2z-3
of convergence.
Sol: Observe that I can be written in the form of a
convergent geometric series as follows:
-1.
£ £ ( 2² )" =
रेस्-डे
n=o
The series expansion is valid only if | 22 | < 1 ⇒ the radius of
convergence of this series. R = 3/₁
is
Some natural questions to pose at this point are: What
happens if we asked to find the power series representation
of a function of that cannot be expressed as a geometric
series? Is there a general method by which we can find
the power series representation of an arbitrary function?
How do we even know that a function admits a power
Series representation ? Now we will partially answer these questions.
Thm 13.1 (Taylor series). Let f(z) be an analytic function on a
domain √ let Z = aεr be some point in J. Assume that a
Circle C of radius r centered at Z=a is contained in J.
Then, there exists a unique power series with radius of
convergence at least r that converges to f(z):
f(z) = = \(^) (a) (z-a)" | f(n)(a)| ≤
f(n)
M
n=o
where
n!
Mis the maximum of (f(z) on the circle 1Z-al=r=
14(z)|
ella z
M = max
1z-al=r
n=2
>
The series on the right is called the Taylor series for I at a.
Rmk For Q=0 the series is called the Maclaurin series.
Namely, of is differentiable arbitrarily many times at a and
this is a unique representation of f as power series on C.
Note: We conclude that if f is once differentiable, then can
be written as power series,
then
we can dedece:
By Prop 12.2: f'(z) = { f(n)(a) (z-a)n-1
n=1
(n-1)!
p(n) (a)
(n-2)!
n
n
и
2
и
1/2 (²) "z" = -2 ²3 tl Z²
n=o
(z-a)n-2
2
and
so on for all ZEC.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 1 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
