PROPERTY 4. E(-2) = zt = (E(z))"'. Problem 18. Prove Property 4.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
100%
#18
1:29
( Safari
RealAnalysis-ISBN-fix...
+ y)?
O!
E(r+ y). O
PROPERTY 3. If m is a positive integer then E(mr) = (E(r))". In particular,
E(m) = (E(1))".
CALCULUS IN THe 17TH AND 18TH CENTURIES
41
Problem 17. Prove Property 3.
PROPERTY 4. E(-x) = x = (E(2))-.
Problem 18. Prove Property 4.
PROPERTY 5. If n is an integer with n+ 0, then E() = VE(1) = (E(1))/"
Problem 19. Prove Property 5.
PROPERTY 6. Ifm and n are integers with n # 0, then E () = (E(1))/".
Problem 20. Prove Property 6.
Definition 2. Let E(1) be denoted by the number e. Using the series e =
E(1) = Eo, we can approrimate e to any degree of accuracy. In particular
e 2.71828.
In light of Property 6, we see that for any rational number r, E(r) = e". Not
only does this give us the series representation e" = Eo r" for any rational
number r, but it gives us a way to define e for irrational values of r as well.
That is, we can define
e = E(r) =
for any real number z.
As an illustration, we now have ev =E (V2)". The expression ev2 is
meaningless if we try to interpret it as one irrational number raised to another.
What does it mean to raise anything to the v2 power? However the series
En (V2)" does seem to have meaning and it can be used to extend the
exponential function to irrational exponents. In fact, defining the exponential
function via thie eorine anewore the oneetion un raiend on nase DT What does
Next
Dashboard
Calendar
To Do
Notifications
Inbox
因
Transcribed Image Text:1:29 ( Safari RealAnalysis-ISBN-fix... + y)? O! E(r+ y). O PROPERTY 3. If m is a positive integer then E(mr) = (E(r))". In particular, E(m) = (E(1))". CALCULUS IN THe 17TH AND 18TH CENTURIES 41 Problem 17. Prove Property 3. PROPERTY 4. E(-x) = x = (E(2))-. Problem 18. Prove Property 4. PROPERTY 5. If n is an integer with n+ 0, then E() = VE(1) = (E(1))/" Problem 19. Prove Property 5. PROPERTY 6. Ifm and n are integers with n # 0, then E () = (E(1))/". Problem 20. Prove Property 6. Definition 2. Let E(1) be denoted by the number e. Using the series e = E(1) = Eo, we can approrimate e to any degree of accuracy. In particular e 2.71828. In light of Property 6, we see that for any rational number r, E(r) = e". Not only does this give us the series representation e" = Eo r" for any rational number r, but it gives us a way to define e for irrational values of r as well. That is, we can define e = E(r) = for any real number z. As an illustration, we now have ev =E (V2)". The expression ev2 is meaningless if we try to interpret it as one irrational number raised to another. What does it mean to raise anything to the v2 power? However the series En (V2)" does seem to have meaning and it can be used to extend the exponential function to irrational exponents. In fact, defining the exponential function via thie eorine anewore the oneetion un raiend on nase DT What does Next Dashboard Calendar To Do Notifications Inbox 因
Expert Solution
Step 1: Given.

Given: E(x)=n=0xnn!=ex

To prove: E(-x)=1E(x)=E(x)-1

steps

Step by step

Solved in 2 steps

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,