...Some Applications of the Residue Theorem∗ Supplementary Lecture Notes MATH 322, Complex Analysis Winter 2005 Pawel Hitczenko Department of Mathematics Drexel University Philadelphia, PA 19104, U.S.A. email: phitczenko@math.drexel.edu ∗I would like to thank Frederick Akalin for pointing out a couple of typos. 1 1 Introduction These notes supplement a freely downloadable book Complex Analysis by George Cain (henceforth referred to as Cain’s notes), that I served as a primary text for an undergraduate level course in complex analysis. Throughout these notes I will make occasional references to results stated in these notes. The aim of my notes is to provide a few examples of applications of the residue theorem. The main goal is to illustrate how this theorem can be used to evaluate various types of integrals of real valued functions of real variable. Following Sec. 10.1 of Cain’s notes, let us recall that if C is a simple, closed contour and f is analytic within the region bounded by C except for finitely many points z0 , z1 , . . . , zk then k f (z)dz = 2πi C j=0 Resz=zj f (z), where Resz=a f (z) is the residue of f at a. 2 2.1 Evaluation of Real-Valued Integrals. Definite integrals involving trigonometric functions 2π We begin by briefly discussing integrals of the form F (sin at, cos bt)dt. 0 (1) Our method is easily adaptable for integrals over a different range, for example between 0 and π or between ±π. Given the form of an integrand...
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...semicircle [pic] 3) the circle [pic] h) If [pic] and C is the arc from z = -1 - i to z = 1 + i along the curve [pic]. i) If [pic] and C is the curve from z = 0 to z = 4+2i given by [pic]. j) Evaluate [pic] along: a) The parabola [pic] [pic] b) Straight line from (0, 3) to (2, 3) and then from (2, 3) to (2, 4) c) A straight line from (0, 3) to (2, 4). EXERCISE-05 1. State Cauchy-Goursat theorem . 2. Verify Cauchy-Goursat theorem for the function [pic], if C is the circle [pic] (b) the circle [pic]. 3. State the Cauchy’s integral formula and Cauchy Residue Theorem. 4. Evaluate by Cauchy’s integral formulae and by Residue theorem (a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic] (f) [pic] 5. Let C denotes the boundary of a square with boundary lines [pic] and [pic] described positively. Evaluate [pic]. 6. Evaluate [pic]...
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...Gunshot Residue Analysis You are provided with cotton balls that were used to swab 2 suspect’s hands for the presence of gunshot residue. Determine, by atomic absorption spectroscopy, if either of the 2 have fired a gun recently. To do this you will first extract the lead residue off of the cotton ball and then you will run the lead samples and a lead positive sample and a blank to determine if there is lead in the residue from the suspect’s hands. Extraction/Sample Prep: 1.) Use tweezers or forceps to remove the cotton ball from its container and place it into a plastic centrifuge tube. Add 10 mL of 1% nitric acid and shake well. Continue shaking occasionally for 10 minutes. 2.) Remove the supernatant (the liquid in the tube) with a transfer pipet and place it into a labeled 25 mL volumetric flask. 3.) Add 1 mL of Ammonium dihydrogen phosphate solution. 4.) Dilute the contents of the flask with 1% nitric acid. 5.) Label the flask(s) as unknowns (1,2). Positive and negative control samples: 1.) Add 1 mL of lead standard to a 25 mL volumetric flask. 2.) Add 1 mL of Ammonium dihydrogen phosphate solution. 3.) Dilute the contents of the flask with 1% nitric acid. 4.) Label the flask as positive control #1 5.) Add 0.5 mL of lead standard to a 25 mL volumetric flask. 6.) Add 1 mL of Ammonium dihydrogen phosphate solution. 7.) Dilute the contents of the flask with 1% nitric acid. 8.) Label the flask as positive control #2 9.) Add 1 mL of Ammonium dihydrogen phosphate solution. 10.)...
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...The officer was asked what was his job regarding this case. He said he looked at the body for physical struggle and did not find any. The prosecution did not have any further questions. The defense started questioning the officer; the defense asked the officer if he did everything the detective on the case asked him to do. The defense then asked if the detective who was assigned to the case asked him if he had looked for gunshot residue. The officer stated that there would be gunshot residue if she were shot. Then the defense asked why did the detective not ask to check for gunshot residue? The officer responded that he did not know. The defense attorney then said this is definitely a case of bad evidence. The prosecutor objected. Judge Sheeran said he would let the comment...
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...paint chips and how are they evaluated? How is soil collected and preserved at a crime scene. What standard/reference samples need to be collected? How is soil examined? What does the scientist look for? Trace evidence is anything left behind at a scene of a crime that can potentially link a suspect and a victim to a mutual location. Examples of trace evidence are hair, fibers, gunshot residue, soil and pollen. Any of these items can be transferred between two or more people, objects or environment with little contact, so during a crime its easily done (Trace, 2012). Analysis of trace material begins with a visual examination of the evidence while Macrophotography is taken to ensure approximate pictures are taken or all aspects of the scene and surrounding evidence. It’s then followed by micro-analysis, a process that allows scientists to investigate the evidence at a microscopic level to analyze substances like chemical residues (Trace, 2012). Other equipment used when analyzing trace materials are electron microscopes- used for the analysis of gunshot residue, spectrometry, spectroscopy and chromatographs- used to analyze small explosives, volatile hydro-carbonic chemicals and vehicular paint samples (Corbis, 2006). To analyze vehicle paint, scientist must breakdown its complex mixture of color. Paint is formed of pigments, modifiers, extenders, and binders. Pigment is what gives paint its color, modifiers control the gloss, extenders add...
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...Back in 2009 in Wampum, Pennsylvania, an eight month old pregnant woman was found shot with a bullet in her head. The search for her killer ended with charging the women’s fiancé’s 11-year-old son with her murder. He is one of the youngest defendants to be charged with homicide and not only was he charged with one homicide but two because of the unborn child’s death as well. The 11-year-old pleaded not guilty to the charges. During the time that more court hearings were taking place, the boy’s family said that he was innocent and should be tried in juvenile court. After he was arrested, they had placed him in the County Jail, but authorities later moved him to a juvenile center because his attorneys argued them that the county jail couldn't accommodate a boy that young. It is rare in the United States to charge a child that young as an adult, the prosecutor in the case said Pennsylvania’ s law left him with little choice. The prosecutor said, “my choice is either to charge him as an adult, or don't charge him… not charging him at all wasn't feasible." It also is rare for a child his age to commit such a violent crime or any violent crime at all for that matter, so this case took a long time to go through trial. The 11-year-old spent a long time in the Adolescent Detention Center, such as long as a birthday and Christmas spent there. The boy’s defense argues that...
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...Renitra Woods CJ370 Unit 9 Assignment July 14, 2014 Footprint and footwear evidence is very valuable. A footprint is the impression that every individual leaves on a surface that has some sort of moisture, it can also be left on a dry surface if the footwear or feet are damp they walk on it. If footprints are undisturbed they can last on a surface from a small amount of time to a long period of time. Footprints should be used as evidence because footprints may be able to assist investigators in proving that an individual committed a crime, or was present at the scene of a crime and could have possibly committed the crime. Footprints are beneficial but not as beneficial as fingerprints or DNA evidence (LaMance, 2000). When investigators look at footprints as evidence they are usually focused on the soles and heel indents, cuts or tread of the shoe itself, and other details such as the type of shoe and the size. One of the first things investigators want to do when collecting a footprint on a scene is take a photograph, document what and where they found the evidence, and create a plaster mold. Footwear is extremely important because it allows examiners to analyze the type of shoe the victim or the perpetrator had on, the size, if there was any damage in cases that may involve the individual wearing footwear such as a high heel shoe as well as the treads of the shoe sole. Preserving and Photographing the evidence is important as well. When investigators approach...
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...People V. Clark Case # 2008-223042-FC Started March 30th Judge: Shalina Kumar Judge Shalina D. Kumar was an associate of the Southfield based law firm of Weiner & Cox, PLC, specializing in complex civil litigation, medical malpractice and wrongful death actions with extensive trial experience. Judge Kumar graduated with distinction from the University of Michigan with a B.A. in psychology and the University of Detroit-Mercy School of Law. http://www.oakgov.com/circuit/judges/kumar-shalina-pro.html Def Attorney: David Rosenburg Pros: Brett Chudler (toughest case in 18 yrs) Charges: Found guilty on all counts 2 Counts assault with intent to murder or assault with intent to do great bodily harm. 3 counts of being in possession of a firearm while committing a felony. 1 count of being a felon in possession of a firearm. Jury: 14 members 8 female 2 black 10am Came into court and only the defendant and his attorneys where in court with 2 sheriffs deputies and Rosenburg was on his computer playing solitary while his co chair is talking about playing card games in college and some other random things while the client is just sitting there. Pros walks in and Rosenburg says they want to do closing statements because the rest of their subpoenaed witnesses did not show up. Shortly after the judge walks in; jury is not present Pros tells judge that the def att should tell his client he has the right to testify Def att wants to read special instructions to the jury about the...
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...Blood Spatter Pattern Analysis Kimberley Kanuch-Brown Everest University Abstract Given that blood spatter analysis is an emergent field, with rapidly occurring developments have significant probative implications for the court system, this project will seek to examine the multi-faceted elements of blood spatter analysis to provide an overview of the field’s different dimensions. Focusing on technical developments, analytical interpretation and court relevance, the project will propose that blood spatter’s analysis as a mainstream element of the CSI toolkit results from the combination of physical sciences and analytical rigor which lies at its core. Beginning with questions of technical and physical science, the project will examine the manner in which blood spatter analysis is increasingly capable of understanding how human usage of different weapons impacts the static nature of a given crime scene. Moving to questions of interpretation, the project will touch upon the manner in which information technology is improving the rigor and caliber of analysis, and thus leading to greater continuity and replicability in blood spatter analysis. Concluding, it will examine the manner in which the court system is now viewing blood spatter analysis with greater heft because of these developments. Blood Spatter Analysis and Science Beginning with questions of science, the work of Randall (2009) demonstrates how even the most esoteric of weapons can be analyzed using blood-spatter...
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...Evidence suggests that the investigation of crimes occurred as far back as prehistoric times when early man took fingerprints by pressing hands and fingers into clay or rock. In Nova Scotia, archaeologists uncovered an ancient drawing of the detailed ridge patterns of a hand. Additionally, the ancient Babylonians also made use of fingerprinting on clay tablets for business transactions and as a means to preserve identification for other official needs. Forensic science continued to evolve, and in 700 B.C., the Chinese began to record thumb prints on clay sculpture and documents despite having no formal classification system was in place. In 250 B.C., an ancient Greek physician, Erasistratus, created the first lie detector test when he noted that his patients' pulse rate appeared to increase when they lied. In 44 B.C., a Roman doctor examined Julius Caesar's body and discovered that of his 23 stab wounds, only one was fatal. The 1100 A.D., crime scene investigation had improved to such a degree that Roman attorney Quintilian proved that blood-covered palm prints were left at a crime scene to frame a blind man for the murder of his mother. Nearly 150 years later, in 1248 AD, the first documentation of medical knowledge aimed at solving crimes was recorded in the book "Hsi Duan Yu" (the Washing Away of Wrong). It detailed the process of distinguishing drowning from strangulation. Over the next 500 years, advancements in crime scene investigation focused on the many aspects...
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...firearm or weapon is personal safety and the safety of anyone who will potentially come in contact with the item of evidence. The investigator then has to preserve the potential evidence the item may reveal. From semiautomatic pistols to shotguns and long rifles each weapon has different mechanical parts and disabled properly before handling. The forensic technician is tasked with rendering all firearms safe so they can be preserved and analyzed at the lab. Firearms are used in many crimes around the world. It is important to know the proper steps in handling firearms in any crime scene investigation. Analysis and comparison of firearms evidence depends on the type used. There are certain rules for collecting firearms, bullets and residue from a crime scene. Each one is different in characteristics from semiautomatic pistols to long rifles and shotguns. These weapons continue to evolve in manufacturing and forensic technicians must stay up to date and educated on new firearm technology. Handguns are most commonly used to commit crimes because they are smaller and can be concealed easily. Long rifles and shotguns, also referred to as shoulder-operated firearms, are used by criminals less frequently. A handgun can consist of a semiautomatic pistol or a revolver. A semiautomatic fires a bullet and automatically reloads another cartridge each time the trigger is pulled. A magazine is the housing of the ammunition (bullets) and can vary in round amount...
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...the responsible person. This is done by carefully documenting the conditions at a crime scene and recognizing all relevant physical evidence. The ability to recognize and properly collect physical evidence is oftentimes critical to both solving and prosecuting violent crimes. It is no exaggeration to say that in the majority of cases, the law enforcement officer who protects and searches a crime scene plays a critical role in determining whether physical evidence will be used in solving or prosecuting violent crimes. Documenting crime scene conditions can include immediately recording transient details such as lighting (on/off), drapes (open/closed), weather, or furniture moved by medical teams. Certain evidence such as shoeprints or gunshot residue is fragile and if not collected immediately can easily be destroyed or lost. The scope of the investigation also extends to considerations of arguments which might be generated in this case (suicide/self-defense) and documenting conditions which would support or refute these arguments. In addition, it is important to be able to recognize what should be present at a scene but is not (victim's vehicle/wallet) and objects which appear to be out of place (ski mask) and might have been left by the assailant. It is also important to determine the full extent of a crime scene. A crime scene is not merely the immediate area where a body is located or where an assailant concentrated his activities but can also encompass a vehicle and access/escape...
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...Crime scene investigation is the meeting point between science, logic and law. Processing a crime scene is and long and tedious process and is the job of crime scene investigators (CSI). CSI has the responsibilities of collecting, documenting, preserving, reconstructing and presenting evidence. It is their job to examine any physical evidence that could remotely shed any light on what happened and who is responsible. There are no typical crime scenes, evidence or investigative approaches and every crime scene should be approached in a systematic sense. According to Robert R. Ogle Jr., a crime scene search is defined as a systematic, methodical search for any physical evidence at a crime scene. Before the job of CSI can begin, there has to be a crime committed. For instance, CSI are on call for murder scene investigations to collect and examine evidence. When CSI is contacted, their job begins. Evidence is a major part in any investigation. Without evidence, a case can grow cold quick. Anything can be evidence and when in a crime scene, everything should be treated as such. Evidence has many different categories: trace, firearm, impression, drug and alcohol, document, and biological fluids. All of these categories of evidence play an important part in finding out valuable information towards solving a case. The most important aspect of evidence collection and preservation is protecting the crime scene. When first approaching a crime scene, first responders should secure the...
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...*Stand Your Ground is a law that is legal in 36% of the United States. The law states that when you are in danger, you do not have to retreat but can instead defend yourself with any means necessary.* 17 year old Trayvon Martin went out on February 26th, 2012 to buy Skittles and iced tea before heading home to watch the local NBA game. George Zimmerman, a resident of Sanford, Florida called 911 to report a “suspicious person”. He was instructed to stay in his SUV and not confront them. Zimmerman disregarded the instructions and began following Martin. Moments later, neighbors reported hearing gunshots. Zimmerman admitted to shooting Martin but claimed it was out of self-defense because Martin had gone for Zimmerman’s gun. When the police...
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...Forensic Chemist The trunk of a stolen car smells of decomposed flesh, a pair of Converse high tops with tiny specs of blood spatter are all alone in the back seat of the same car, and a half smoked cigarette rests softly on a freshly mowed lawn next to the driveway. These are clues of a crime scene. Each clue adds up to the story of something terrible. A crime has taken place, and law enforcement is on the scene. Whether it is a detective, county Sheriff or the FBI, the investigators will rely on their forensic teams to link all the pieces of the puzzle that don’t quite fit yet. A Forensic Chemist can make them fit by scientifically analyzing the evidence. Chemistry, biology, materials science, and genetics to analyze clue found at the scene of the crime, on the victims or in the bodies of the bad guys. Forensic Chemists go into a case with many unknow pieces of the crime scene they need to analyze to determine the nature of each sample. Most Forensic Chemists work in a lab. It is rare for private labs to do this kind of work so most of the time these labs are associated with Local, State, or Federal law enforcement agencies. From local Medical Examiner’s labs to state of the art FBI labs, Forensics Chemists often provide the strongest evidence in court against the defendants. They have many different types of test and methods they use to figure out what the samples mean. Each crime scene brings new types of clues and samples so a Forensic Chemist must always...
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