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Show that I ;r U>l Log( I - i l = - In 2 - -i. - 2 4 (a)

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

Solution for problem 3.21 Chapter Chapter 3

Complex Variables and Applications | 9th Edition

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

Complex Variables and Applications | 9th Edition

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Problem 3.21

Show that I ;r U>l Log( I - i l = - In 2 - -i. - 2 4 (a) loge= I+ 211;ri (11 = 0. 1. +2 .... ): (bl logi = (111 + ~) ;ri (11=0. 1. +2 .... ): (c) log(-1 + J};) = ln2 + 2 (11 + }) ;ri (11 = 0. +I. 2 .... ).

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Life 103 Notes *adapted from the lecture notes of Dr. Tanya Dewey* Animal Diversity and Evolution • Themes of Biology -Organization-structure and function, emergent properties, reductionism -Information-information is transferred between generations through genes (heredity) -Energy and Matter-all organisms require energy and matter, resources are often limited, look for trade-offs -Interactions-interactions are important at all levels of organization (cell to organisms to communities to the biosphere) -Evolution-nothing in biology makes sense except in the light of evolution • Themes in Animal Diversity -Trends in evolution -Tradeoffs and constraints -Different perspectives and ways of life • What is science -Asking testable questions, formulated as hypotheses -Using evidence to answer those questions -Employing parsimony-simple explanations are preferred -Biology is a science -Science is a way of approaching problems that is applicable to nearly every aspect of your life • Eukaryan Diversity -Protists -Plants -Fungi -Animals • What are animals -Metazoans -zoo or zoa=animals ~Ex.) Zoology • How well do we know animals -<1million named species -Estimates up to 7.8 million total -86% of species on land yet to be discovered -91% of marine species yet to be discovered -Even in very common places, there are still many more species to be discovered -Most species on this planet are animals -70% of animals are insects -Animals have been around for a long time, but relative to every other species, they’re relatively recent • Ediacaran Origin -Animals evolved, including extant taxa and extinct forms -Only a few animal phyla evolved • Cambrian Explosion -Oldest fossils of half of extant animal phyla -Most major animal body plans evolve -Almost every other animal phyla that didn’t evolve in Ediacaran evolved in Cambrian • What are animals =Metazoans -Multicellular -Ingestive heterotrophs ~Do not produce own food; take food into body and digest internally -Move under own volition at some point in life ~At some point they were motile -Lack cell walls, have structural proteins (extracellular matrix) -Unique, specialized cells: nerve and muscle (except sponges) ~Most animals have these cells, whereas they aren’t found in any other species -Sexual reproduction -2n (diploid) dominant -Flagellated sperm, non-motile egg -Most have larval stage -Cells are organized into tissues -Conserved genes control development (Hox genes) -Zygote undergoes cleavage, forms blastula, gastrulation • Eumetazoa=true tissues -Doesn’t include sponges because they don’t have true tissues -Metazoa includes sponges, but eumetazoa doesn’t • Themes in animal evolution (see textbook figure 32.11) -Origin of multicellularity -Origin of bilateral symmetry, cephalization, and the nervous system -Origin of embryonic tissue layers -Origin of a coelom -Origin of protostome and deuterostome development -Origin of segmentation • Origin of multicellularity -Choanoflagellates are the unicellular sister group to animals (see textbook figure 32.3) -Evolved at the origin of animals (Metazoa; includes sponges and all other animals) • Origin of bilateral symmetry, cephalization (formation of a head), and the nervous system -Types of symmetry ~Bilateral symmetry (Ex. Beetle) ~Radial symmetry (Ex. Coral polyp) ~No symmetry (Ex. Sponge) ~Pentaradial symmetry/pentamerism (echinoderms, except their larvae have bilateral symmetry) -Bilateral symmetry means you can have a head = cephalization ~Having a head gives the animal directionality (front and a back) ~Having a head gives the animal the ability to eat/be a predator -Concentrate sensory apparatus and nervous system at head -Evolved at the Bilateria origin (has bilateral symmetry if ancestor is Bilateria, Ctenophora and Cnidaria have radial symmetry, and sponges have no symmetry) -Genetic mechanisms responsible for bilateral symmetry and cephalization are shared (homologous) • Origin of embryonic tissue layers -Diploblastic=2 tissue layers (Endoderm and Ectoderm) -Triploblastic=3 tissue layers (Endoderm, Mesoderm, and Ectoderm) -Species with Bilateria ancestor are triploblastic, while Ctenophora and Cnidaria are diploblastic • Origin of a coelom -Animals are tubular-tubes within tubes -Coelom- a fluid-filled body cavity between the inner and outer tubes; disengages gut from outer layers, space for nutrients to move, forms basis of hydrostatic skeleton -3 body plans (see textbook figure 32.9) ~Acoelomate=no coelom ~Coelomate=has coelom ~Pseudocoelomate=false coelom (not a true coelom) -Pseudocoelomate has the space touching both the mesoderm and endoderm, while a true coelom only touches the mesoderm -Cannot say the coelomate evolved at a specific point of the tree because the body plans are randomly distributed ~Diploblastic and sponges don’t have coeloms ~Body plans scattered among Bilateria species • Origin of protostome and deuterostome development -“Stoma”=opening or mouth -Protostome-the mouth is formed before the anus ~”First mouth” -Deuterostome-the anus is formed before the mouth ~”Second mouth -The first opening in protostomes is the mouth, while the first opening in deuterostomes is the anus -Deuterostomes evolved at Deuterostomia -Protosomes, or “Spiralia,” evolved at Lophotrochozoa and Ecdysozoa • Origin of segmentation -Not all animal groups show segmentation -Common genes called Hox genes control segmentation -Suggesting flexible response through evolutionary history -Why segmentation ~Permits specialization-genetic duplication releases copies to be modified for new purposes-look for this theme in vertebrate evolution Animal Taxa: Invertebrates I (Diversity) • Invertebrate Diversity (see textbook figure 33.3) -Morphological and molecular data are combined to understand relationships among animal phyla -Molecular (DNA) data has revolutionized our understanding • “Big 9” animal phyla -Porifera -Cnidaria -Platyhelminthes -Mollusca -Annelida -Nematoda -Arthropoda -Echinodermata -Chordata • *Porifera (sponges) -5,000-10,000 species -Filter feeding -No symmetry -Sessile -Mostly marine • Ctenophora (comb jellies) -100-150 species -Predatory -Radially symmetrical -Motile (via cilia) -Entirely marine • *Cnidaria (jellyfish, corals, anemones) -Over 10,000 species -Predatory or filter feeding -Radially symmetrical -Both motile and sessile -Mostly marine • Placozoa (“flat animals”) -1 to a few species -Detritivore -Radially symmetrical -Motile (via flagella) -Entirely marine • Acoela (acoel flatworms) -Approximately 400 species -Predatory -Bilaterally symmetrical -Motile (via cilia) -Mostly marine • Rotifera (wheel animals) -Filter feeding -Bilaterally symmetrical -Motile -Mostly freshwater • Acanthocephala (spiny headed worms) -1150 species -Parasitic -Bilaterally symmetrical -Motile -Parasitic (freshwater) • Cycliophora -1-2 species -Discovered in 1995, found on the mouthparts of lobsters -Parasitic or commensal -Symmetry is not clear -Motile -Marine • Gastrotricha (hairybacks) -790 species -Detritivores -Motile -Marine and freshwater • Gnathostomulida (jaw worms) -100 species -Detritivore -Motile (via cilia) -Marine • *Platyhelminthes (flatworms) -25,000 species -Predators or parasites -Motile -Moist habitats • Entoprocta (“anus inside”) -150 species -Filter feeding -Sessile and colonial -Mostly marine • *Mollusca (mollusks) -85,000 species -More varied forms than any other phylum -Predatory, filter feeding, detritivores -Both motile and sessile -Marine, freshwater, terrestrial • Sipunculida (peanut worms) -320 species -Filter feeding -Motile -Marine • Brachiopoda (lamp shells) -330 species -Filter feeding -Sessile -Marine • Phoronida (horseshoe worms) -25 species -Filter feeding -Sessile -Marine • Ectoprocta/Bryzoa (moss animals “anus outside”) -4,000 species -Filter feeding -Sessile -Mostly marine • Nemertea (ribbon worms) -1,800 species -Predatory and parasitic -Motile -Mostly marine, some freshwater and terrestrial • *Annelida (segmented worms) -22,000 species -Predatory, filter feeding, detritivores, sanguivores -Both motile and sessile -Marine, freshwater, terrestrial • Priapulida (penis worm) -16 species -Detritivore -Motile -Marine • Kinorhyncha (mud dragons) -Detritivore/predatory -Motile -Marina • Loricifera -120 species -Discovered in 1983 in anoxic, deep sea brine -No mitochondria -Feeding style unknown -Sessile -Marine • Nematomorpha (horsehair worms) -2,000 species -Parasitic -Motile -Freshwater or moist habitats • *Nematoda (roundworms) -15,000 species or up to 1 million -Half parasitic, half free living -Motile -Nearly all habitats and all elevations • Chaetognatha (arrow worms) -120 species -Predatory -Motile -Marine planktonic • Tardigrada (water bears) -1,150 species -Predatory -Motile -Moist habitats, extreme environments • Onychophora (velvet worms) -180 species -Predatory -Motile -Terrestrial • *Arthropoda (arthropods) -Many millions -Every feeding style imaginable -Motile -Marine, freshwater, terrestrial • What are the patterns you observe in the diversity of animal phyla -Most phyla live in marine habitats, however, terrestrial phyla tend to have more overall diversity

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

This full solution covers the following key subjects: . This expansive textbook survival guide covers 12 chapters, and 771 solutions. This textbook survival guide was created for the textbook: Complex Variables and Applications, edition: 9. The answer to “Show that I ;r U>l Log( I - i l = - In 2 - -i. - 2 4 (a) loge= I+ 211;ri (11 = 0. 1. +2 .... ): (bl logi = (111 + ~) ;ri (11=0. 1. +2 .... ): (c) log(-1 + J};) = ln2 + 2 (11 + }) ;ri (11 = 0. +I. 2 .... ).” is broken down into a number of easy to follow steps, and 61 words. Complex Variables and Applications was written by and is associated to the ISBN: 9780073383170. Since the solution to 3.21 from Chapter 3 chapter was answered, more than 239 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 3.21 from chapter: Chapter 3 was answered by , our top Math solution expert on 12/23/17, 04:39PM.

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