# on 06-Dec-2023 (Wed)

#### Annotation 7597067996428

 The relations between the solution of an ODE and the solution of the ODE's auxiliary equation Depending on the solution of the auxiliary equation, the solution of the ODE is different: $$\displaystyle m=\frac{-b\pm\Delta}{2a}, \Delta\equiv b^{2}-4ac$$ 1. $$\Delta>0\Rightarrow$$ 2 real roots: $$m_{1},m_{2}$$, both $$y_{1}=e^{m_{1}x},y_{2}=e^{m_{2}x}$$ works. So the general solution is $$y_{general}=Ae^{m_{1}x}+Be^{m_{2}x}$$ - Alternative form: Because of the Euler's Equation in $$\mathbb{C}$$, $$y(x)=e^{px}[\tilde{A}\cosh(qx)+\tilde{B}\sinh(qx)]$$, where $$p,q$$ are $$\displaystyle m=\underbrace{- \frac{b}{2a}}_p\pm \underbrace{\frac{\sqrt{\Delta}}{2a}}_{q}$$. - Why we need this form? When the initial condition includes $$y(0)$$, the alt form would be much easier 2. $$\Delta<0\Rightarrow$$ 2 complex roots: $$m_{1},m_{2}\in\mathbb{C}$$, so the general solution is $$y(x)=Ce^{m_{1}x}+De^{m_{2}x}$$ - Alternative form: Because of the Euler's Equation, $$y(x)=e^{m_{r}x}[\tilde{C}\cos(m_{i}x)+\tilde{D}\sin(m_{i}x)]$$, where $$m_{1,2}=m_{r}\pm im_{i}$$. 3. $$\Delta=0\Rightarrow$$ one $$m$$ only! so the general solution is $$y(x)=Ae^{mx}+Bxe^{mx}=(A+Bx)e^{mx}$$

#### Flashcard 7597070617868

Question

Depending on the solution of the auxiliary equation, the solution of the corresponding 2nd Linear & homogeneous ODE is different: $$\displaystyle m=\frac{-b\pm\sqrt{\Delta}}{2a}, \Delta\equiv b^{2}-4ac$$

[Answer the normal solutions and their alternative form]

1. $$\Delta>0\Rightarrow$$ 2 real roots: $$m_{1},m_{2}$$, both $$y_{1}=e^{m_{1}x},y_{2}=e^{m_{2}x}$$ works. So the general solution is $$y_{general}=Ae^{m_{1}x}+Be^{m_{2}x}$$
- Alternative form: Because of the Euler's Equation in $$\mathbb{C}$$, $$y(x)=e^{px}[\tilde{A}\cosh(qx)+\tilde{B}\sinh(qx)]$$, where $$p,q$$ are $$\displaystyle m=\underbrace{- \frac{b}{2a}}_p\pm \underbrace{\frac{\sqrt{\Delta}}{2a}}_{q}$$.
- Why we need this form? When the initial condition includes $$y(0)$$, the alt form would be much easier
2. $$\Delta<0\Rightarrow$$ 2 complex roots: $$m_{1},m_{2}\in\mathbb{C}$$, so the general solution is $$y(x)=Ce^{m_{1}x}+De^{m_{2}x}$$
- Alternative form: Because of the Euler's Equation, $$y(x)=e^{m_{r}x}[\tilde{C}\cos(m_{i}x)+\tilde{D}\sin(m_{i}x)]$$, where $$m_{1,2}=m_{r}\pm im_{i}$$.
3. $$\Delta=0\Rightarrow$$ one $$m$$ only! so the general solution is $$y(x)=Ae^{mx}+Bxe^{mx}=(A+Bx)e^{mx}$$

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The relations between the solution of an ODE and the solution of the ODE's auxiliary equation
Depending on the solution of the auxiliary equation, the solution of the ODE is different: $$m=\frac{-b\pm\Delta}{2a}, \Delta\equiv b^{2}-4ac$$ 1. $$\Delta>0\Rightarrow$$ 2 real roots: $$m_{1},m_{2}$$, both $$y_{1}=e^{m_{1}x},y_{2}=e^{m_{2}x}$$ works. So the general solution is $$y_{general}=Ae^{m_{1}x}+Be^{m_{2}x}$$ - Alternative form: Because of the Euler's Equation in $$\mathbb{C}$$, $$y(x)=e^{px}[\tilde{A}\cosh(qx)+\tilde{B}\sinh(qx)]$$, where $$p,q$$ are $$\displaystyle m=\underbrace{- \frac{b}{2a}}_p\pm \underbrace{\frac{\sqrt{\Delta}}{2a}}_{q}$$. - Why we need this form? When the initial condition includes $$y(0)$$, the alt form would be much easier 2. $$\Delta<0\Rightarrow$$ 2 complex roots: $$m_{1},m_{2}\in\mathbb{C}$$, so the general solution is $$y(x)=Ce^{m_{1}x}+De^{m_{2}x}$$ - Alternative form: Because of the Euler's Equation, $$y(x)=e^{m_{r}x}[\tilde{C}\cosh(m_{i}x)+\tilde{D}\sinh(m_{i}x)]$$, where $$m_{1,2}=m_{r}\pm im_{i}$$. 3. $$\Delta=0\Rightarrow$$ one $$m$$ only! so the general solution is $$y(x)=Ae^{mx}+Bxe^{mx}=(A+Bx)e^{mx}$$

#### Annotation 7604059901196

 Conditions and forces that dominate in actual fact the modern world have not attained any coherent intellectual expression of themselves. We live, as is so often remarked, in astate of divided allegiance. In outward activities and current enjoyments, we are frenetically absorbed in mun- dane affairs in ways which, if they were formulated for in- tellectual acceptance, would be repudiated as low and un- worthy. We give our emotional and theoretical assent to prin- ciples and creeds which are no longer actively operative in life. We have retained enough of the older tradition to recognize that aphilosophy which formulated what, on the whole and in the mass, we are most concerned with, would be intolerably materialistic in character. On the other hand, we are not pre- pared, either intellectually or morally, to frame such aphiloso- phy of the interests and activities that actually dominate our lives as would elevate them to aplane of truly liberal and humane significance. We are unable to show that the ideals, values and meanings which the philosophy we nominally hold places in another world, are capable of characterizing in a concrete form, with some measure of security, the world in which we live, that of our actual experience. On this account any sincere empirical philosophy that holds to the possibility of the latter alternative must be prophetic rather than descriptive.

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#### Annotation 7604061474060

 here is adistinction between hypotheses generated in that seclusion from observable fact which renders them fantasies, and hypotheses that are projections of the possi- bilities of facts already in existence and capable of report. There is adifference between the imaginative speculations that recognize no law except their own dialectic consistency, and those which rest on an observable movement of events, and which foresee these events carried to alimit by the force of their own movement. There is adifference between support by argument from arbitrarily assumed premises, and an argument which sets forth the implications of propositions resting upon facts already vitally significant.

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#### Annotation 7604063833356

 If, accordingly, it can be shown that the actual procedures by which the most authentic and de- pendable knowledge is attained have completely surrendered the separation of knowing and doing jif it can be shown that overtly executed operations of interaction are requisite to obtain the knowledge called scientific, the chief fortress of the classic philosophical tradition crumbles into dust. With this destruction disappears also the reason for which some objects, as fixed in themselves, out of and above the course of human experience and its consequences, have been set in opposition to the temporal and concrete world in which we live.

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#### Annotation 7604065406220

 the application of natural science, through the medium of inventions and technologies, is the finally controlling and characteristic fact of modern life. That western civilization is increasingly in- dustrial in character is acommonplace} it should be an equally familiar fact that this industrialization is the direct fruit of the growth of the experimental method of knowing. The effects of this industrialization in politics, social arrangements, communication and intercourse, in work and play, in the de- termination of the locus of influence, power and prestige, are characteristic marks of present experience in the concrete. They are the ultimate source of that waning of the effective influence of older beliefs that has been alluded to. They also provide the reason why aphilosophy which merely reflected and re- ported the chief features of the existing situation as if they were final, without regard to what they may become, would be so repulsively materialistic. Both the positive fact that our actual life is more and more determined by the results of physical science, and the negative fact that these results are so largely an obstacle to framing aphilosophy consonant with

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#### Annotation 7604066979084

 Knowledge in its full and valid sense is possible only of the immutable, the fixed} that alone answers the quest for cer- tainty. With regard to changing things, only surmise and opinion are possible, just as practically these are the source of peril. To ascientific man, in terms of what he does in in- quiry, the notion of anatural science which should turn its back upon the changes of things, upon events, is simply incom- prehensible. What he is interested in knowing, in understand- ing, are precisely the changes that go onj they set his problems, and problems are solved when changes are interconnected with one another. Constants and relative invariants figure, but they are relations between changes, not the constituents of ahigher realm of Being. With this modification with respect to the object comes one in the structure and content of "experience." Instead of there being afixed difference between it and some- thing higher rational thought there is adifference between two kinds of experience jone which is occupied with uncon- trolled change and one concerned with directed and regulated change. And this difference, while fundamentally important, does not mark afixed division.

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#### Annotation 7604068551948

 The central and outstanding fact is that the change in the method of knowing, due to the scientific revolution begun in the sixteenth and seventeenth centuries, has been accompanied by arevolution in the attitude of man toward natural occur- rences and their interactions. This transformation means, as was intimated earlier, acomplete reversal in the traditional relationship of knowledge and action. Science advances by adopting the instruments and doings of directed practice, and the knowledge thus gained becomes ameans of the develop- ment of arts which bring nature still further into actual and potential service of human purposes and valuations.

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#### Annotation 7604070124812

 o notion could be further away from the fact than the some- what sedulously cultivated idea that the difference between ancient and modern science is that the former had no respect for perception and relied exclusively upon speculation. In fact, the Greeks were keenly sensitive to natural objects and were keen observers. The trouble lay not in substitution of theoriz- ing from the outset for the material of perception, but in that they took the latter "as is"; they made no attempt to modify it radically before undertaking thinking and theorizing about it.

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#### Annotation 7604071697676

 hese statements would be misunderstood if they were taken to imply an allegation that in ancient science sense gives knowledge, while modern science excludes the material of sense} such an idea inverts the facts. But ancient science accepted the material of sense-material on its face, and then organized it, as it naturally and originally stood, by operations of logical definition, classification into species and syllogistic subsumption. Men either had no instruments and appliances for modifying the ordinary objects of observation, for analyz- ing them into their elements and giving them new forms and arrangements, or they failed to use those which they had. Thus in content, or subject-matter, the conclusions of Greek science (which persisted till the scientific revolution of the seventeenth century), were much closer to the objects of everyday experi- ence than are the objects of present scientific thought. It is not meant that the Greeks had more respect for the function of per- ception through the senses than has modern science

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#### Annotation 7604073270540

 judged from present practice, they had altogether too mucfy respect for the material of direct, unanalyzed sense-perception.

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#### Annotation 7604074843404

 e passage from ordinary perception to scientific knowledge did not therefore demand the introduction of actual, overt and observed changes into the material of sense perception. Modern science, with its changes in the subject-matter of direct perception effected by the use of apparatus, gets away not from observed material as such, but from the qualitative characteristics of things as they are originally and "naturally" observed. It may thus be fairly asserted that the "categories" of Greek description and explanation of natural phenomena were esthetic in character jfor perception of the esthetic sort is interested in things in their immediate qualitative traits. The logical features they depended upon to confer scientific form upon the material of observation were harmony, proportion or measure, symmetry: these constitute the "logos" that renders phenomena capable of report in rational discourse. In virtue of these properties, superimposed upon phenomena but thought to be elicited from them, natural objects are knowable. Thus the Greeks employed thinking not as ameans of changing given objects of observation so as to get at the conditions and effects of their occurrence, but to impose upon them certain

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#### Annotation 7604076416268

 Craftsmen, architects, sculptors, gymnasts, poets had taken raw material and con- verted it into finished forms marked by symmetry and pro- portion jthey accomplished this task without the prior dis- integrative reduction which characterizes modern making in the factory. Greek thinkers performed alike task for nature as awhole. Instead, however, of employing the material tools of the crafts, they depended upon thought alone. They bor- rowed the form provided them in Greek art in abstraction from its material appliances. They aimed at constructing out of nature, as observed, an artistic whole for the eye of the soul to behold. Thus for science nature was acosmos. It was com- posed, but it was not acomposite of elements. That is, it was a qualitative whole, awhole as is adrama, astatue or atemple, in virtue of apervading and dominant qualitative unity jit was not an aggregate of homogenous units externally arranged in different modes. Design was the form and pattern in- trinsically characteristic of things in their fixed kinds, not some- thing first formed in adesigning mind and then imposed from without.

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#### Annotation 7604077989132

 Iknow of no one thing more significant for an under- standing of Greek science than Aristotle's treatment of quantity as an accident, that is, as something which can vary within limits (set by the inherent essence and measure, logos) of a thing without affecting its nature. When we think of the Cartesian definition of quantity as the essence of matter, we appreciate that an intellectual revolution has taken place: a radical change in point of view and not just the product of more, and more accurately stated, information, but achange involving surrender of the esthetic character of the object. Contrast the place occupied in modern science by relations with the Aristotelian illustrations of their nature namely, distinc- tions of more and less, greater and smaller, etc. For the point of Aristotle's treatment is that relations, like quantity, are indifferent to the essence or nature of the object, and hence are of no final account for scientific knowledge. This conception is thoroughly appropriate to an esthetic point of view, wherein that which is internally complete and self-sufficing is the all- important consideration.

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#### Annotation 7604079561996

 ovement was aterm covering all sorts of qualitative alterations, such as warm things becoming cold, growth from embryo to adult form, etc. It was never conceived of as merely motion, i.e., change of position in a homogeneous space. When we speak of amusical movement, or apolitical movement, we come close to the sense attached to the idea in ancient science: aseries of changes tending to complete or perfect aqualitative whole and fulfill an end.

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#### Annotation 7604081397004

 There was no conflict with ideas about values, because the qualities belonging to objects of science are values jthey are the things we enjoy and prize. Throughout nature as aqualitative whole there is a hierarchy of forms from those of lower value to those of higher. The revolution in science effectively initiated by Galileo consisted precisely in the abolition of qualities as traits of scientific objects as such. From this elimination proceeded just that conflict and need of reconciliation between the scientific properties of the real and those which give moral authority. Therefore to apprehend what the new astronomy and physics did for human beliefs, we have to place it in its contrast with the older natural science in which the qualities possessed by objects of scientific knowledge were precisely the same as those possessed by works of art, the properties which are one with beauty and with all that is admirable.

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#### Annotation 7604082969868

 mass or inertia became the scientific definition or stable co-efficient of matter, in complete indifference to the qualitative differentiations of wet-dry, hot-cold, which were henceforth things to be explained by means of mass and motion, not funda- mental explanatory principles

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#### Annotation 7604084542732

 alileo's conclusions were absolutely fatal to the traditional conception that all bodies in motion come naturally to rest because of their own intrinsic tendency to fulfill an inherent nature. T

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#### Annotation 7604086115596

 it substitutes data for objects. (It is not meant that this outcome is the whole effect of the experimental method; that as we saw at the outset is complex; but that the first effect as far as stripping away qualities is concerned is of this nature.) That Greek science operated with objects in the sense of the stars, rocks, trees, rain, warm and cold days of ordinary experience is evident enough. What is signified by saying that the first effect of experimenta- tion was to reduce these things from the status of objects to that of data may not be so clear.* By data is signified subject- matter for further interpretation; something to be thought about. Objects are finalities; they are complete, finished; they call for thought only in the way of definition, classification, logical arrangement, subsumption in syllogisms, etc. But data signify "material to serve"; they are indications, evidence, signs, clues to and of something still to be reached; they are intermediate, not ultimate; means, not finalities.

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#### Annotation 7604087688460

 The subject-matter which had been taken as satisfying the demands of knowledge, as the material with which to frame solutions, became something which set problems. Hot and cold, wet and dry, light and heavy, instead of being self-evident mat- ters with which to explain phenomena, were things to be investi- gated; they were "effects," not causal principles; they set question marks instead of supplying answers. The differences between the earth, the region of the planets, and the heavenly ether, instead of supplying ultimate principles which could be used to mark off and classify things, were something to be explained and to bring under identical principles.

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#### Annotation 7604089261324

 he remarkable difference between the attitude which ac- cepts the objects of ordinary perception, use and enjoyment as final, as culminations of natural processes and that which takes them as starting points for reflection and investigation, is one which reaches far beyond the technicalities of science. It marks arevolution in the whole spirit of life, in the entire attitude taken toward whatever is found in existence. When the things which exist around us, which we touch, see, hear and taste are regarded as interrogations for which an answer must be sought (and must be sought by means of deliberate introduction of changes till they are reshaped into something different), nature as it already exists ceases to be something which must be accepted and submitted to, endured or en- joyed, just as it is. It is now something to be modified, to be intentionally controlled. It is material to act upon so as to transform it into new objects which better answer our needs. Nature as it exists at any particular time is achallenge, rather than acompletion jit provides possible starting points and op- portunities rather than final ends.

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#### Annotation 7604090834188

 he esthetic attitude is of necessity directed to what is already there jto what is finished, com- plete. The attitude of control looks to the future, to pro- duction.

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#### Annotation 7604092407052

 hile the abolition of fixed tendencies toward definite ends has been mourned by many as if it involved adespiritualization of na- ture, it is in fact aprecondition of the projection of new ends and of the possibility of realizing them through intentional activity. Objects which are not fixed goals of nature and which have no inherent defining forms become candidates for receiving new qualities} means for serving new purposes. Until natural objects were denuded of determinate ends which were regarded as the proper outcome of the intrinsic tendency of nature's own operations, nature could not become aplastic material of human desires and purposes.

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#### Annotation 7604093979916

 It is because of injection of an irrelevant philosophy into interpretation of the conclusions of science that the latter are thought to eliminate qualities and values from nature. Thus is created the standing problem of modern philosophy: the relation of science to the things we prize and love and which have authority in the direc- tion of conduct. The same injection, in treating the results of mathematical-mechanistic science as adefinition of natural real- ity in its own intrinsic nature, accounts for the antagonism shown to naturalism, and for the feeling that it is the busi- ness of philosophy to demonstrate the being of arealm beyond nature, one not subject to the conditions which mark all natural objects. Drop the conception that knowledge is knowledge only when it is adisclosure and definition of the properties of fixed and antecedent reality jinterpret the aim and test of knowing by what happens in the actual procedures of scientific inquiry, and the supposed need and problem vanish

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#### Annotation 7604095552780

 instead of accepting the qualities and values the ends and forms of this world as providing the objects of knowledge, subject to their being given acertain logical arrangement, experimental inquiry treats them as of- fering achallenge to thought. They are the materials of problems not of solutions. They are to be known, rather than objects of knowledge. The first step in knowing is to locate the problems which need solution. This step is per- formed by altering obvious and given qualities. These are effects} they are things to be understood, and they are under- stood in terms of their generation. The search for "efficient causes" instead of for final causes,

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#### Annotation 7604097125644

 It is unnecessary that knowledge should be concerned with existence as it is directly experienced in its concrete qualities. Direct experiencing itself takes care of that matter. What science is concerned with is the ha^enmg of these experienced things. For its purpose, therefore, they are happenings, events. Its aim is to discover the conditions and consequences of their happening. And this discovery can take place only by modify- ing the given qualities in such ways that relations become mani- fest. We shall see later that these relations constitute the proper objects of science as such.

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#### Annotation 7604098960652

 Only when the older theory of knowledge and metaphysics is retained, is science thought to inform us that nature in its true reality is but an interplay of masses in motion, without sound, color, or any quality of enjoyment and use. What science actually does is to show that any natural object we please may be treated in terms of relations upon which its occurrence depends, or as an event, and that by so treating it

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#### Annotation 7604100533516

 As long, for example, as water is taken to be just the thing which we directly expedience it to be, we can put it to afew direct uses, such as drinking, washing, etc. Beyond heating it there was little that could be done purposefully to change its properties. When, however, water is treated not as the glisten- ing, rippling object with the variety of qualities that delight the eye, ear, and palate, but as something symbolized by FLO, something from which these qualities are completely absent, it becomes amenable to all sorts of other modes of control and adapted to other uses. Similarly, when steam and ice are no longer treated as what they are in their qualitative differences from one another in direct experience, but as homogeneous molecules moving at measured velocities through specified dis- tances, differential qualities that were barriers to effective regu- lations, as long as they were taken as finalities, are done away with. Asingle way of acting with respect to them in spite of their differences is indicated. This mode of action is capable of extension to other bodies, in principle to any bodies irre- spective of qualitative differences of solid, liquid and gaseous, provided they are given alike mathematical formulation. Thus all sorts of modes of expansion and contraction, of re- frigeration and evaporation, of production and regulation of explosive power, become possible.

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#### Annotation 7604102106380

 Water as an object of science, as HUO with all the other scientific propositions which can be made about it, is not arival for position in real being with the water we see and use. It is, because of experimental operations, an added instrumentality of multiplied controls and uses of the real things of everyday experience.

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#### Annotation 7604103679244

 These preconceptions are the assumption that knowledge has auniquely privileged position as amode of access to reality in comparison with other modes of experience, and that as such it is superior to practical activity. Both of these ideas were formulated in a period when knowing was regarded as something which could be effected exclusively by means of the rational powers of mind. The development of scientific inquiry with its complete dependence upon experimentation has proved the profound error of the latter position. Is it not time to revise the philo- sophical conceptions which are founded on abelief now proved to be false? The sum and substance of the present argument is that if we frame our conception of knowledge on the experimental model, we find that it is away of operating upon and with the things of ordinary experience so that we can frame our ideas of them in terms of their interactions with one another, instead of in terms of the qualities they directly present, and that thereby our control of them, our

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#### Annotation 7604105252108

 nowing is itself amode of practical action and is the way of interaction by which other natural interactions become subject to direction. Such is the significance of the experimental method as far as we have as yet traced its course.

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#### Annotation 7604107087116

 The pattern supplied by scientific know- ing shows that in this one field at least it is possible for ex- perience, in becoming genuinely experimental, to develop its own regulative ideas and standards. Not only this, but in addi- tion the progress of knowledge of nature has become secure and steady only because of this transformation. The conclusion is agood omen for the possibility of achieving in larger, more humane and liberal fields asimilar transformation, so that a philosophy of experience may be empirical without either being false to actual experience or being compelled to explain away the values dearest to the heart of man.

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#### Annotation 7604140379404

 Definition of Taylor Series The Taylor series of a infinitely differentiable function $$f(x)\in\mathbb{C}$$ at a single point $$a\in\mathbb{C}$$ is defined as: $${\displaystyle f(a)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}}$$ where $$f^{(n)}(a)$$ denotes the $$n$$th derivative of $$f$$ evaluated at the point $$a$$. - The derivative of order zero of $$f$$ is defined to be $$f$$ itself and $$(x − a)^{0}$$ and $$0!$$ are both defined to be 1.

#### Flashcard 7604143787276

Question
The Taylor series of a infinitely differentiable function $$f(x)\in\mathbb{C}$$ at a single point $$a\in\mathbb{C}$$ is defined as: [...]

$${\displaystyle f(a)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}}$$

where $$f^{(n)}(a)$$ denotes the $$n$$th derivative of $$f$$ evaluated at the point $$a$$.
- The derivative of order zero of $$f$$ is defined to be $$f$$ itself and $$(x − a)^{0}$$ and $$0!$$ are both defined to be 1.

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Definition of Taylor Series
The Taylor series of a infinitely differentiable function $$f(x)\in\mathbb{C}$$ at a single point $$a\in\mathbb{C}$$ is defined as: $${\displaystyle f(a)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}}$$ where $$f^{(n)}(a)$$ denotes the $$n$$th derivative of $$f$$ evaluated at the point $$a$$. - The derivative of order zero of $$f$$ is defined to be $$f$$ itself and $$(x − a)^{0}$$ and $$0!$$ are both defined to be 1.

#### Annotation 7604154273036

 Definition of Fourier series The Fourier series of a 2L-periodic function $$f(x)$$ is defined as: $$\displaystyle f(x)=\frac{a_{0}}{2}+\sum\limits^{\infty}_{n=1}a_{n}\cos\left(\frac{n\pi x}{L}\right)+b_{n}\sin\left(\frac{n\pi x}{L}\right)$$ where - $$\displaystyle a_{0}=\frac{1}{L}\int^{L}_{-L}f(x)dx\equiv \left\langle1,f\right\rangle$$. - $$\displaystyle a_{n}=\frac{1}{L}\int^{L}_{-L}f(x)\cos\left(\frac{n\pi x}{L}\right)dx\equiv \frac{\left\langle\cos_{n},f\right\rangle}{\left\langle\cos_{n},\cos_{n}\right\rangle}$$. - $$\displaystyle b_{n}=\frac{1}{L}\int^{L}_{-L}f(x)\sin\left(\frac{n\pi x}{L}\right)dx\equiv \frac{\left\langle\sin_{n},f\right\rangle}{\left\langle\sin_{n},\sin_{n}\right\rangle}$$.

#### Flashcard 7604157680908

Question
The Fourier series of a 2L-periodic function $$f(x)$$ is defined as: [...]

$$\displaystyle f(x)=\frac{a_{0}}{2}+\sum\limits^{\infty}_{n=1}a_{n}\cos\left(\frac{n\pi x}{L}\right)+b_{n}\sin\left(\frac{n\pi x}{L}\right)$$

where
- $$\displaystyle a_{0}=\frac{1}{L}\int^{L}_{-L}f(x)dx\equiv \left\langle1,f\right\rangle$$.
- $$\displaystyle a_{n}=\frac{1}{L}\int^{L}_{-L}f(x)\cos\left(\frac{n\pi x}{L}\right)dx\equiv \frac{\left\langle\cos_{n},f\right\rangle}{\left\langle\cos_{n},\cos_{n}\right\rangle}$$.
- $$\displaystyle b_{n}=\frac{1}{L}\int^{L}_{-L}f(x)\sin\left(\frac{n\pi x}{L}\right)dx\equiv \frac{\left\langle\sin_{n},f\right\rangle}{\left\langle\sin_{n},\sin_{n}\right\rangle}$$.

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Definition of Fourier series
The Fourier series of a 2L-periodic function $$f(x)$$ is defined as: $$\displaystyle f(x)=\frac{a_{0}}{2}+\sum\limits^{\infty}_{n=1}a_{n}\cos\left(\frac{n\pi x}{L}\right)+b_{n}\sin\left(\frac{n\pi x}{L}\right)$$ where - $$\displaystyle a_{0}=\frac{1}{L}\int^{L}_{-L}f(x)dx\equiv \left\langle1,f\right\rangle$$. - $$\displaystyle a_{n}=\frac{1}{L}\int^{L}_{-L}f(x)\cos\left(\frac{n\pi x}{L}\right)dx\equiv \frac{\left\langle\cos_{n},f\right\rangle}{\left\langle\cos_{n},\cos_{n}\right\rangle}$$. - $$\displaystyle b_{n}=\frac{1}{L}\int^{L}_{-L}f(x)\sin\left(\frac{n\pi x}{L}\right)dx\equiv \frac{\left\langle\sin_{n},f\right\rangle}{\left\langle\sin_{n},\sin_{n}\right\rangle}$$.

#### Annotation 7604160040204

 Important Inner Products For specific weight funciton $$w(x)=1$$ and domain $$x\in(-L,L)$$, then - $$\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \cos\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \cos\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}$$ - $$\displaystyle\left\langle \sin\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\sin\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}$$ - $$\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=0$$

#### Flashcard 7604161875212

Question
For specific weight funciton $$w(x)=1$$ and domain $$x\in(-L,L)$$, then: [Important inner products]
- $$\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \cos\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \cos\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}$$
- $$\displaystyle\left\langle \sin\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\sin\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}$$
- $$\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=0$$
For specific weight funciton $$w(x)=1$$ and domain $$x\in(-L,L)$$, then - $$\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \cos\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \cos\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}$$ - $$\displaystyle\left\langle \sin\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\sin\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}$$ - $$\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=0$$