Prove: If ƒ € R(a) on [a,b] and g € R(a) on [a, b] , then fg e R(a) on [a,b]

Given if f∈Rα on a,b and g∈Rα on a,b. We have to prove that fg∈Rα on a,b. Since, f∈Rα on a,b then f…

Answered: Prove: If ƒ e R(a) on [a,b] and g e…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

Prove: If ƒ € R(a) on [a,b] and g € R(a) on [a, b] , then fg e R(a) on [a,b]

Expert Solution

Step 1

Given if $f \in R (α)$ on $[a, b]$ and $g \in R (α)$ on $[a, b]$ .

We have to prove that $f g \in R (α)$ on $[a, b]$ .

Since, $f \in R (α)$ on $[a, b]$ then $f$ is bounded on $[a, b]$

Since, $g \in R (α)$ on $[a, b]$ then $g$ is bounded on $[a, b]$

Therefore, $f g$ is bounded on $[a, b]$ .

Now, $f g$ can be written as follows

$f g = \frac{1}{2} {(f + g)}^{2} - \frac{1}{2} f^{2} - \frac{1}{2} g^{2}$

Theorem:- If $f \in R (α)$ and $g \in R (α)$ on $[a, b]$ then ${(f + g)}^{2}$ , $f^{2}$ and $g^{2}$ are all integrable on $[a, b]$ .

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

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,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

Prove: If ƒ e R(a) on [a,b] and g e R(a) on [a, b] , then fg e R(a) on [a,b]