Define the branch of the logarithm log-/2 given by Recall that log-/2 is defined for all nonzero z € C, but it is only analytic on =C\ {-iy: y ≥ 0), with derivative for z EN. z Let 0
Define the branch of the logarithm log-/2 given by Recall that log-/2 is defined for all nonzero z € C, but it is only analytic on =C\ {-iy: y ≥ 0), with derivative for z EN. z Let 0
Chapter6: Exponential And Logarithmic Functions
Section6.8: Fitting Exponential Models To Data
Problem 15SE: What is the y -intercept on the graph of the logistic model given in the previous exercise?
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answer c and d
![Define the branch of the logarithm log-/2 given by
Recall that log-/2 is defined for all nonzero z € C, but it is only analytic on
=C\ {-iy: y ≥ 0), with derivative for z EN.
z
Let 0 <r < 1< R. Define Yr(t) = Ret for t = [0, ]. Let TR be the closed curve
given by [r, R] UYRU[-R, -r] U-% (where -% is the same curve % but traced
in the opposite direction).
(a) Use Cauchy's Residue Theorem to evaluate
log-/22
Sar
(b) Show in detail exactly one of the following.
log-T/2z
(2² + 1)²
dz0 as R→∞
dz0 as r0+
●
log-/22= In |2|+i arg-/22, arg-/22 E
(c) Write
log-/22
(22 + 1)²
•La
DO In x
log-/22
(√ng + √(1-1-1) ) ( 2² + 1)² d
dz
as a single integral with respect to the positive real variable z.
(d) From the previous results, deduce that
and
5
2+1)²
(22
z € (-1,77).
dz.
π
dx = -
50%
1
(x² + 1)²
dx =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0cfb37ca-2c4c-4cc0-8df9-2ef95f1d2a36%2Fc6dd12f1-d8e5-4f0c-a2ef-acff0a9c95bd%2F2mwlmff_processed.png&w=3840&q=75)
Transcribed Image Text:Define the branch of the logarithm log-/2 given by
Recall that log-/2 is defined for all nonzero z € C, but it is only analytic on
=C\ {-iy: y ≥ 0), with derivative for z EN.
z
Let 0 <r < 1< R. Define Yr(t) = Ret for t = [0, ]. Let TR be the closed curve
given by [r, R] UYRU[-R, -r] U-% (where -% is the same curve % but traced
in the opposite direction).
(a) Use Cauchy's Residue Theorem to evaluate
log-/22
Sar
(b) Show in detail exactly one of the following.
log-T/2z
(2² + 1)²
dz0 as R→∞
dz0 as r0+
●
log-/22= In |2|+i arg-/22, arg-/22 E
(c) Write
log-/22
(22 + 1)²
•La
DO In x
log-/22
(√ng + √(1-1-1) ) ( 2² + 1)² d
dz
as a single integral with respect to the positive real variable z.
(d) From the previous results, deduce that
and
5
2+1)²
(22
z € (-1,77).
dz.
π
dx = -
50%
1
(x² + 1)²
dx =
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