×
Log in to StudySoup
Get Full Access to Complex Variables And Applications - 9 Edition - Chapter Chapter 12 - Problem 12.2
Join StudySoup for FREE
Get Full Access to Complex Variables And Applications - 9 Edition - Chapter Chapter 12 - Problem 12.2

Already have an account? Login here
×
Reset your password

Let F he a picccv ... isc continuous funclion (Sec. 42) of

Complex Variables and Applications | 9th Edition | ISBN: 9780073383170 | Authors: James Ward Brown ISBN: 9780073383170 169

Solution for problem 12.2 Chapter Chapter 12

Complex Variables and Applications | 9th Edition

  • Textbook Solutions
  • 2901 Step-by-step solutions solved by professors and subject experts
  • Get 24/7 help from StudySoup virtual teaching assistants
Complex Variables and Applications | 9th Edition | ISBN: 9780073383170 | Authors: James Ward Brown

Complex Variables and Applications | 9th Edition

4 5 1 419 Reviews
17
3
Problem 12.2

Let F he a picccv ... isc continuous funclion (Sec. 42) of 8 on chc inlcrval 0 ~ A ::::: 2JT. The Poisson integral tra11sform of F is defined in tcm1s of the Poisson kernel P(ro. r. - 8 ). introduced in Sec. 134, hy means of the equation ( I ) I 12:-r U (r. H) = ~ P(ro.r. - H) F< - H ) I F ( ) - F ( 8) I d . 2n o For convenience. we let F be extended periodically. with period 2;rr. so that the integrand here is perilxlic in wilh that same period. Also. we agree that 0 < r < r0 because of the nature of the limit to be established. Next. we observe that since F is continuous al H. there is a small positive number ex such that (4) f; IFll - F(8)1 < ., whenever l-8l~ex. Assume now that I - HI ~ a and write (5) where I 1 111u /1 (r) = - P(r0 . r. - H) IF() - F({Jl) d. 2;rr II U h(r) = ~ { uil:r P(r0 r. - H) [F() - F(8)) d. _;rr Jn.1 u The fact that P is a positive function (Sec. 134). together with the first of inequalities (4) just above and properly (_{). Sec. 134. of that function. enables us to write I {111a l/1 (r)I ~ 'l;rr Jo u P(ro. r. - H) IF() - F(8)1 d f; f.l:r F. < - P(r0 r. - 8) d = -. 4;rr . 0 2 As for the integral /2 (r). one can sec from Fig. 192 in Sec. 134 that the denominator Is - :-f in expression (8) for P(ro. r. - H) in that section has a (positive) minimum value 111 as the argument of s v:.u-ics over the closed interval 8 + (X ~ ~ e - ex + 2;rr . So. if M denotes an upper bound of the piecewise continuous function I Fl) - F(Hll on the in le rval 0 ~ ~ 2;rr. it follows th al (r ~ - r 2 )M 2Mr11 21Wro F l/:~(r)I ~ 2:r < --(r0 - r) < -- 15 = - 2 :r Ill 111 111 ., 422 ll\TEGRAL fooRMt:L..\S or: TllE PoL~sol'\ TYPE whenever t(i - r < ,5 where ( 6) 111 F: D=--. 4M ro Finally. the results in the t\vo preceding paragraphs tell us that /-~ IU(r.8) - FI:::; l/i(r)I + lh (II= 0. J. 2 ... . ). iT () I ;2:r b,1 = - F(

Step-by-Step Solution:
Step 1 of 3

LIB – 120 – Introduction to Psychology Chapter 1: The Science of Psychology Vocabulary  Behavioral Perspective: A research perspective whose major explanatory focus is how external environmental events condition observable behavior. (3)  Biological Perspective: A research perspective whose major explanatory focus is how the brain, nervous system, and other physiological mechanisms produce behavior and mental processes. (2)  Case Study: A descriptive method in which the researcher studies an individual in depth over an extended period of time. (9)  Cognitive Perspective: A research perspective whose major explanatory focus is how the mental processes such as perception, memory, and problem solving, work and impact behavior. (2)  Control Group: In an experiment, the group not exposed to the independent variable. (18)  Correlation Coefficient: A statistic that tells us the type and the strength of the relationship between two variables. The sign of the coefficient ( + or - ) indicates the type of correlation – positive or negative, respectively. The absolute value of the coefficient ( 0.0 – 1.0 ) represents the strength of the correlation, with 1.0 being the maximum strength. (12)  Correlation Study: A research study in which two variables are measured to determine if they are related (How well either one predicts the other). (11)  Dependent Variable: In an experiment, a variable that is hypothesized to be affected by the independent variable and thus is measured by the experimenter. (17)  Descriptive Methods: Research methods whose main purpose is to provide objective and detailed descriptions of behavioral and mental processes. (8)  Descriptive Statistics: Statistics that describe the results of a research study in a concise fashion. (24)  Double-Blind Procedure: A control measure in an experiment in which neither the experimenters nor the participants know which participants are in the experimental and control groups. (21)  Experiment: A research method in which the researcher manipulated one or more independent variables and measures their effect on one or more dependent variables while controlling other potentially relevant variables. (17)  Experimental Group: In an experiment, the group exposed to the independent variable. (17)  Frequency Distribution: A depiction, in a table or figure, of the number of participants (frequency) receiving each score for a variable. (24)  Hindsight Bias: (I-Knew-It-All-Along Phenomenon) The tendency, after learning about and outcome, to be over confident in one’s ability to have predicted it. (6)  Independent Variable: In an experiment, the variable that is a hypothesized cause and thus is manipulated by the experimenter. (17)  Inferential Statistical Analyses: Statistical analyses that allow researchers to draw conclusions about the results of a study by determining the probability that the results are due to random variation (chance). The results are statistically significant if this probability is .05 or less. (20)  Left-Skewed Distribution: An asymmetric frequency distribution in which there are some unusually low scores that distort the mean to be less than the median. (29)  Mean: The numerical averages of a distribution of scores. (25)  Median: The score positioned in the middle of a distribution of scores when all of the scores are arranged from lowest to highest. (25)  Meta-Analysis: A statistical technique that combines the results of a large number of studies on one experimental question into one analysis to arrive at an overall conclusion. (22)  Mode: The most frequently occurring score in a distribution of scores. (25)  Naturalistic Observation: A descriptive research method in which the behavior of interest is observed in its natural setting, and the researchers does not intervene in the behavior being observed. (8).  Negative Correlation: An inverse relationship between two variables. (12)  Nocebo Effect: A negative placebo effect due to the expectation of adverse consequences from receiving treatment. (19)  Normal Distribution: A frequency distribution that is shaped like a bell. About 68% of the scores fall within 1 standard deviation of the mean, about 95 % within 2 standard deviations of the mean, and over 99% within 3 standard deviations of the mean. (27)  Operational Definition: A description of the operations or procedures that a researcher uses to manipulate or measure a variable. (18)  Participant Observation: A descriptive research method in which the observed becomes part of the group being observed. (9)  Percentile Rank: The percentage of scores below a specific score in the distribution of scores. (28)  Placebo Effect: Improvement due to the expectation of improving because of receiving treatment. (18)  Placebo Group: A control group of participants who believe they are receiving treatment, but who are only receiving a placebo. (19)  Placebo: An inactive pill or a treatment that has no known effects. (18)  Population: The entire group of people that a researcher is studying. (10)  Positive Correlation: A direct relationship between two variables. (12)  Psychology: The science of behavior and mental processes. (1)  Random Assignment: A control measure in which participants are randomly assigned to groups in order to equalize participant characteristics across the various groups in an experiment. (16)  Random Sampling: A sampling technique that obtains a representative sample of a population by ensuring that each individual is a population has an equal opportunity to be in the sample. (11)  Range: The difference between the highest and lowest scores in a distribution of scores. (25)  Right-Skewed Distribution: An asymmetric frequency distribution in which there are some unusually high scores that distort the mean to be greater than the median. (28)  Sample: The subset of a population that actually participates in a research study. (11)  Scatterplot: A visual depiction of correlational data in which each data point represents the scores on the two variables for each participant. (13)  Sociocultural Perspective: A research perspective whose major explanatory focus is how external environmental events condition observable behavior. (4)  Standard Deviation: The average extent that the scores vary from the mean for a distribution of scores. (26)  Survey Research: A descriptive research method in which the researchers uses questionnaires and interviews to collect information about the behavior, beliefs, and attitudes in particular groups of people. (10)  Third-Variable Problem: An explanation of a correlation between two variables in terms of another (third) variable that could possibly be responsible for the observed relationship between the two variables. (15)  Variable: Any factor that can take on more than one value. (11) LIB – 120 – Introduction to Psychology Chapter 1: The Science of Psychology Chapter Outline  Psychology is a science, not just a mental health profession.  Psychology is the science of behavior and mental processes. THE FOUR MAJOR RESEARCH PERSPECTIVES  There are 4 major research perspectives  Biological  Cognitive  Behavioral  Sociocultural  It’s important to understand that all of the perspectives are complementary to one another.  They are simply pieces of a jigsaw puzzle, fitting together to form a picture and a complete explanation of our behavior and mental processes. Major Goals of Psychologists  To explain human behavior and mental processes. o Internal and External Factors can provide emphasis to the 4 major perspectives. o Biological and Cognitive perspective focus on causes that stem from within us (internal Factors) o Behavioral and sociocultural perspective focus on causes that stem from outside us (external Factors) Perspectives Emphasizing Internal Factors  Biological Perspective - Our physiological hardware (especially the brain and nervous system) is viewed as the major determiner of behavior and mental processing In Contrast:  Cognitive Perspective – the major explanatory focus is on how mental processes, such as perception, memory, and problem solving, work and impact our behavior.  The Biological Perspective  Biological psychologists look for causes with our physiology, our genetics, and human evolution.  They study the involvement of the various parts of the brain and nervous system on our behavior and mental processes.  The brain not only controls vision, but it is also the control center for almost all of our behavior and mental processing.  The Cognitive Perspective  Cognitive psychologists study all aspects of cognitive processing from perception to the higher-level processes, such as problem solving and reasoning. Perspectives Emphasizing External Factors  Both behavioral perspective and sociocultural perspective focus on external factors in explaining human behavior and mental processing.  The behavioral perspective emphasizes conditioning of out behavior by environmental events, and there is more emphasis on explaining observable behavior than on unobservable mental processes  The sociocultural perspective also emphasizes the influence of the external environment.  The Behavioral Perspective  We behave as we do because of our past history of conditioning by our environment  There are two types of conditioning  Classical – or – Pavlovian Conditioning  Operant Conditioning  Classical Conditioning  Example: When you give a dog food in its mouth and play a specific tone together, overtime you will find that if you just play the tone the dog will begin to salivate in anticipation of receiving food due to being conditioned over time  Operant Conditioning  Involves the relationship between our behavior and its environment  Example: If you are reinforced of praised for a behavior, it will most likely cause the behavior to increase, if we are punished on the other hand, the behavior is likely to decrease and cease.  Environmental events condition our behavior and are the causes of it.  The Sociocultural Perspective  Focuses on the impact of other people (individuals and groups) and our cultural surroundings on our behavioral and mental processing  We are social animals; therefore other people are important to use and thus greatly affect what we do and how we think.  Bystander Effect:  The more bystanders there are to an event such as a murder, raping, attack, or even robbery, the less likely someone is to do anything about it. This is because EVERYONE believes that someone else has already acted by doing the right thing and had called the police or alerted the services needed to provide aid.  Example: Kitty Genovese Murder in 1964  38 reportedly saw/heard but nothing was done until it was too late.  Developmental Psychology  The scientific study of human development across the life span.  One must be aware of the hindsight bias – the tendency after learning about an outcome to be over confident in ones ability to have predicted it. Research Methods Used by Psychologists  All Psychologists, regardless of their perspective use the same research methods.  These methods fall into three categories:  Descriptive  Correlational  Environmental  The experimental method is used most often because it allows the researcher to explore cause-effect relationships  Researchers sometimes are unable to conduct experiments due to ethical restrictions  In situations where experiments can not be carried out, psychologists may employ other methods – descriptive and correlational.  Relationships = Correlations  Descriptive Methods  There are 3 types of descriptive methods  Observational Techniques  Case Studies  Survey Research  The main purpose of all three methods is to provide objective and detailed descriptions of behavior and mental processes  These methods only allow researchers to speculate about cause-effect relationships – to develop hypotheses about casual relationships  Such hypotheses must then be tested in experiments Observational Techniques  Observational techniques exactly reflect their name  The researcher directly observes the behavior of interest  Such observations can be done in a laboratory; for example, children’s behavior can be observed using a one- way mirror.  Behavior in a laboratory may not be natural. This is why researchers often use naturalistic observation.  A descriptive research method in which behavior is observed in its natural setting, without the researcher intervening in the behavior being observed.  Researchers use naturalistic observation when they are interested in how humans or other animals behave in their natural environments.  Observational techniques do have a major problem however. The observer may influence or change the behavior of those being observed. This is why observers must remain unobtrusive as possible, so that the results wont be contaminated by their presence  To observe this possible shortcoming, researchers use participant observation.  When the observer becomes part of the group being observed.  Sometimes naturalistic observation studies start out with unobtrusive observation and end up as participant observation.  Example: Dian Fossey’s study on gorillas turned into a participant observation when the group she was studying finally accepted her.  Case Studies  Detailed observation is also involved in a case study.  In a case study, the researcher studies an individual in depth over an extended period of time.  The researcher attempts to learn as much as possible about the individual being studied.  A life history for the individual is developed, and data for a variety of tests are collected.  Case studies are commonly used in clinical settings when patients are suffering from specific deficits or problems  The main goal of a case study is to gather information that will help in the treatment of the patient.  Case studies are specific to an individual and can not be generalized to a population.  The data accrued from a case study however may be used to help researchers develop hypotheses that then can be tested in experimental research.  Survey Research  The last descriptive method.  Survey researchers use questionnaires and interviews to collect information about the behavior, beliefs, and attitudes of particular groups of people. (It is assumed that participants are willing and answer the questions accurately)  Biased answers may be given due to the wording, order, and structure of the survey questions.  Social Desirability Bias – Our tendency to respond in socially approved ways that may not reflect what we actually think or do.  A necessity in survey research is surveying a representative sample of the relevant population – the entire group being studied.  A researcher only tests a sample – or a subset of people in a population participating in a study.  For these sample data to be meaningful, the sample has to be representative of the larger relevant population, if you don’t have a representative sample, then generalization of the survey findings to the population is not possible.  In random sampling, each individual in the population has an equal opportunity of being in the sample.  Think of selecting names out of the hat, where each name has an equal opportunity of being selected.  Correlational Studies  In a correlational study, two variables are measured to determine if they are related.  A Variable is any factor that can take on more than one value. For example, age, height, GPA, and intelligence test scores are variables.  In conducting a correlational study, the researcher first gets a representative sample of the relevant population. Next the researcher takes 2 measurements on the sample. For example, the researcher could measure a person’s height and weight.  The Correlation Coefficient  To see if the variables are related, the researcher calculates a statistic called the correlation coefficient - A statistic that tells us the type and strength of the relationship between two variables.  Correlation coefficients range from -1.0 to 1.0.  The sign – or + tells us the type of relationship, negative, or positive.  A Positive correlation indicates a direct relationship between two variables – low scores on one variable tend to be paired with low scores on the other variable, and visa vera, (high scores with high scores)  A Negative Correlation is an inverse relationship between two variables. (Low scores on one variable paired with a high score on the other variable)  The strength of the correlation coefficient is based off the absolute value. 0.0 -1.0  Zeros and absolute values close to zero indicate no relationship between the two variables. As the value increases to 1.0, the strength increases.  As the strength increases, researchers can predict the relationship with more accuracy.  Scatterplot  A good way to understand the predictability of a coefficient is to examine a scatterplot – a visual depiction of correlation data.  Each point represents the scores of the two variables.  The Third-Variable Problem  Strong correlations give us excellent predictability, but they do not allow us to draw cause-effect conclusions about the relationships between variables.  Correlation is necessary, but not sufficient for causation to exist Only data from well-controlled experiments allow us to draw conclusions. Example: Consider the negative correlation between self- esteem and depression. As self esteem decreases, depression increases, but we cannot conclude that low self- esteem causes depression.  Third-Variable Problem – another variable may be responsible for the relationship observed between two variables. (Stress could cause depression or self- esteem, biological variables, ect.) Experimental Research  The key aspect of experimental research is that the researcher controls the experimental setting, the only factor that varies is what the researcher manipulates.  It is this tight control of the experimental setting that allows the researcher to make cause-effect statements about the experimental results.  This is done in two steps 1. The experimenter controls for all possible influence of third variables by making sure that they are held constant across all of the groups or conditions in the experiment. 2. The experimenter controls for any possible influences due to the individual characteristics of the participants, such as intelligence, motivation, and memory by using random assignment.  Random Assignment – randomly assigning the participants to groups in an experiment in order to equalize participant characteristics across the various groups in the experiment.  Differences between random sampling and random assignment  Random sampling is a technique in which a sample of participants that is representative of a population is obtained.  Designing an Experiment  The researcher begins with a hypothesis about the cause- effect relationship between two variables.  One of the two variables is assumed to be the cause, and the other one is the one affected.  The Independent variable – is the hypothesized cause, and the experimenter manipulates it.  The Dependent variable – is the variable that is hypothesized to be affected by the independent variable, and thus is measured by the experimenter.  In an experiment, the researcher manipulates the independent variable and measures its effect on the dependent variable while controlling other potentially relevant variables.  Example: There are two groups, one group is exposed to the independent variable, and the other is not. The group exposed to the independent variable is known as the experimental group, and the one not exposed is known as the control group.  The independent and dependent variables in an experiment must be operationally defined.  An Operational Definition is a description of the operations or procedures the researcher uses to manipulate or measure a variable.  We also need a control for what is called the placebo effect - improvement due to the expectation of improving because of receiving treatment. The treatment involved in the placebo effect, however, only involves receiving a placebo – an inactive pill or a treatment that has no known effects.  This effect can arise not only from a conscious belief, but a subconscious belief as well.  Saying that the “pill” is expensive, that is it found in a big brand name can be linked to helping cause patients finding themselves “better”  The placebo has a “evil Twin” the Nocebo effect, where expectation of a negative outcome due to treatment leads to adverse effects.  Sometimes referred to as the negative placebo effect due to its potential to harm Placebo (latin for) - I will please Nocebo (Latin for ) - I will Harm  Researchers add a placebo group to control for the possible placebo effect. A Placebo Group – is a group of participants who believe they are receiving treatment, but they are not, they simply get a placebo medicine.  Inferential Statistical Analyses – statistical analyses that allow researchers to draw conclusions about the results of their studies.  Such analyses tell the researcher the probability that the results of their experiment are due to random variation (chance)  The Double-Blind Procedure  Neither the experimenters nor the participants know which participants are in the experimental and control groups. (Referred to as blind because neither the experimenters or participants know the group assignments)  It is not unusual for participants to not know which groups they are in; this is especially critical for the placebo group participants.  As an experimenter increases the number of values of an independent variable, the number or independent variables, or the number of dependent variables, the possible gain in knowledge about the relationship between the variables also increases.  Researchers have a statistical technique called Meta Analysis – that combines the results for a large number of studies on one experimental question into one analysis to arrive at an overall conclusion.  Because meta-analysis used the results from multiple experiments centered around the same question, the conclusion that is arrived at is considered much stronger evidence than the results of an individual study in answering the experimental question. How To Understand Research Results  To understand the results of a completed experiment, statistics is needed to be used.  There is two types of statistics – descriptive and inferential.  Descriptive Statistics-  Statistics used to describe the data of a research study in a concise fashion.  The correlation coefficient is a descriptive statistic that allows us to describe the results of a correlational study precisely.  For experimental findings, we need two types of descriptive statistis to summarize our data, Measures of variability and Measures of central tendency, in addition, a researcher often constructs a frequency distribution for the data.  A Frequency distribution – depicts, in a table or a graph, the number of participants receiving each score for a variable. The bell curve, or normal distribution is the most famous frequency distribution.  In an experiment, the data set consists of the measures scores on the dependent variable for the sample participants. o Measures of Central Tendency  Defines a “typical” score for a distribution of scores. There are three measures of central tendency (three ways to define the “typical” score) – mean, median, mode.  Mean- the numerical average for a distribution of scores.  Median – the score positioned in the middle of the distribution of scores.  Mode – the most frequently occurring score in the distribution of scores. o Measures of Variability  You need to determine the variability between the scores as well, not just the typical distribution.  There are two measures of variability – range, and distribution.  Range – the difference between the highest and lowest scores in the distribution  The measure of variability that is used most often is standard deviation. Standard deviation is the average extent that the scores vary from the mean of the distribution. (how spread out are the scores)  If the scores do vary greatly from the mean, the standard deviation will be small. o Frequency Distributions  Organizes the data in a score distribution so that we know the frequency of each score.  Tells us how often each score occurred.  For many human traits, the frequency takes on the shape of a bell curve.  Statisticians call this bell-shaped frequency distribution the normal distribution. o Normal Distribution  There are two main aspects in normal distribution.  The mean, median and the mode are all equal because the normal distribution is symmetric about its center.  The percentage of scores falling within a certain number of standard deviations of the mean is set.  About 68% fall within 1 stdev. 95% within 2 stdev., and 99% within 3 stdev. Of the mean.  These percentages are what give the normal distribution is bell shape. The percentages hold regardless of the size of the standard deviation.  The percentages of scores and the numbers of standard deviations from the mean always have the same relationship in a normal distribution. This allows you to compute the percentile rank. This is the percentage of scores below a specific score in a distribution of scores. o Skewed Distributions  In addition to the normal distribution, two other types of frequency distributions are important. They are known as skewed distributions, which are frequency distributions in asymmetric shapes.  There are two major types of skewed distributions, the right and left skewed distributions.  Right Skewed Distribution – is a frequency distribution in which ther are some unusually high scores.  Left Skewed Distribution – is a frequency in which there are some unusually low scores.  Unusually high or low scores distort a mean.  Skewed distributions are important to understand because various aspects of everyday life such as medical trends are often skewed (mortality rates for various diseases)

Step 2 of 3

Chapter Chapter 12, Problem 12.2 is Solved
Step 3 of 3

Textbook: Complex Variables and Applications
Edition: 9
Author: James Ward Brown
ISBN: 9780073383170

Other solutions

People also purchased

Related chapters

Unlock Textbook Solution

Enter your email below to unlock your verified solution to:

Let F he a picccv ... isc continuous funclion (Sec. 42) of