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1 E(z)E(y) %3D 1! 1 - y)² + O! = E(x + y). O PROPERTY 3. If m is a positive integer then E(mz) = (E(z))". In particular, E(m) = (E(1))" . CALCULUS IN THE 17TH AND 18TH CENTURIES 41 Problem 17. Prove Property 3.

1 E(z)E(y) %3D 1! 1 - y)² + O! = E(x + y). O PROPERTY 3. If m is a positive integer then E(mz) = (E(z))". In particular, E(m) = (E(1))" . CALCULUS IN THE 17TH AND 18TH CENTURIES 41 Problem 17. Prove Property 3.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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
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