Explanation Suppose lim s → α + i β [ ( s − α ) 2 + β 2 ] P ( s ) Q ( s ) = A + i B ... ( 1 ) Take the conjugate on both sides in the above equation. lim s → α − i β [ ( s − α ) 2 + β 2 ] P ( s ) Q ( s ) = lim s → α + i β [ ( s − α ) 2 + β 2 ] P ( s ) Q ( s ) ¯ = A − i B Expand the rational term P ( s ) Q ( s ) as follows. P ( s ) Q ( s ) = A + i B s − ( α + i β ) + A − i B s − ( α − i β ) + R ˜ ( s ) = ( A + i B ) ( s − ( α − i β ) ) + ( A − i B ) ( s − ( α + i β ) ) [ s − ( α + i β ) ] [ s − ( α − i β ) ] + R ˜ ( s ) = 2 A ( s − α ) + 2 B β ( s − α ) 2 + β 2 + R ˜ ( s ) ... ( 2 ) Replace 2 A by A and 2 B by B in the equation ( 2 ) . P ( s ) Q ( s ) = A ( s − α ) + B β ( s − α ) 2 + β 2 + R ˜ ( s ) ... ( 3 ) Claim that the following equation is true. lim s → α + i β [ ( s − α ) 2 + β 2 ] P ( s ) Q ( s ) = B β + i A β ... ( 4 ) Now, consider the right hand side of the above equation

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

7th Edition

ISBN: 9780321977182

Author: Nagle, R. Kent, Saff, Edward B., Snider, Arthur David

Publisher: PEARSON

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Chapter 7.4, Problem 42E

To determine

To prove:

The portion of the partial fraction expansion of P(s)/Q(s) corresponding to (s−α)2+β2 is A(s−α)+βB(s−α)2+β2.

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Chapter 7.4, Problem 41E

Chapter 7.4, Problem 43E

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You may need to use the appropriate appendix table or technology to answer this question. You are given the following information obtained from a random sample of 4 observations. 24 48 31 57 You want to determine whether or not the mean of the population from which this sample was taken is significantly different from 49. (Assume the population is normally distributed.) (a) State the null and the alternative hypotheses. (Enter != for ≠ as needed.) H0: Ha: (b) Determine the test statistic. (Round your answer to three decimal places.) (c) Determine the p-value, and at the 5% level of significance, test to determine whether or not the mean of the population is significantly different from 49. Find the p-value. (Round your answer to four decimal places.) p-value = State your conclusion. Reject H0. There is insufficient evidence to conclude that the mean of the population is different from 49.Do not reject H0. There is sufficient evidence to conclude that the…

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Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

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To prove: The portion of the partial fraction expansion of P ( s ) / Q ( s ) corresponding to ( s − α ) 2 + β 2 is A ( s − α ) + β B ( s − α ) 2 + β 2 .

Solution Summary: The author explains the portion of the partial tion expansion of P(s) corresponding to left (s-alpha)2+beta

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To prove:

The portion of the partial fraction expansion of P(s)/Q(s) corresponding to (s−α)2+β2 is A(s−α)+βB(s−α)2+β2.

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Chapter 7 Solutions

To prove: The portion of the partial fraction expansion of P ( s ) / Q ( s ) corresponding to ( s − α ) 2 + β 2 is A ( s − α ) + β B ( s − α ) 2 + β 2 .

Solution Summary: The author explains the portion of the partial tion expansion of P(s) corresponding to left (s-alpha)2+beta

Concept explainers

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To prove: The portion of the partial fraction expansion of P(s)/Q(s) corresponding to (s−α)2+β2 is A(s−α)+βB(s−α)2+β2.

Want to see the full answer?

Chapter 7 Solutions

To prove:

The portion of the partial fraction expansion of P(s)/Q(s) corresponding to (s−α)2+β2 is A(s−α)+βB(s−α)2+β2.