If we assume that
The fluid flow is continuous and the fluid velocity is continuous
The fluid flow is two dimensional
The fluid flow is steady
then we can represent the velocity of the fluid at each point by a continuous complex function whose domain is the region occupied by the fluid.
In fact we must havewithand
To find the flow in a direction at angleanticlockwise from the real axis (since anticlockwise is taken to be in the positive sense) relative to the real axis we can rotateby the anglethen take the real part, so thatwhereis the complex conjugate ofand similarly
We are interested in the real part ofand this is the same if we take the complex conjugate ofThe conjugate ofis in some ways more useful than
Theorem
A steady two dimensional fluid flow with continuous velocity functionon a regionis a model flow ( circulation and flux free, so that the fluid has constant density) if and only the conjugate velocity function bar v is analytic on
Any functionanalytic on a regioncan be considered as the conjugate velocity function of a continuous velocity functionon
Example: If (is analytic onthenis the model flow velocity function.
]]>Definition Letandbe analytic functions whose domains are the regionsand respectively.andare direct analytic continuations of each other if there is a regionsuch thatfor
Example:is only defined forbut we may writeon the region defined bybutis defined on the wider regionis an analytic continuation of
Example:is normally defined on the regionso that
We in fact only need Log to be one to one on the complex plane so that the inverse is defined. We can instead define Log-1 z on so thatforand we have extendedto be defined on
]]>a) is a compact subset of
b) is symmetric under reflection in the real axis
c) meets the real axis in the interval
d) has no holes it it
Proof
Foreach iterate of 0,defines a polynomial function of
and so on. In generalfor
To prove a) define the setfor n=1,2,3,... so thatand so on.
andso that M is bounded .
Each setis closed because it has complementwhich is open. In fact iffor somethen this inequality must also hold for allin some open disc with centreby the continuity of the functionIt follows thatmust be closed because ifthenfor someand so some open disc with centremust lie outsideand hence outsideThis proves thatis closed and bounded, or compact.
To prove b), notice that becauseis a polynomial inwith real coefficients,forbutif and only ifby symmetry ofin the real axis.
To prove c), noteis disconnected forandsoforand
To prove d) we can proveis connected so that each pair of points incan be joined by a path inConsider the setWe can show that each point incan be joined to a point ofby a path in
Suppose in fact thatis a point ofwhich cannot be joined toin this way. Define
sinceandis open because ifcan be joined tothen so can any point of any open disc inwith centreandis connected because pairs of points incan be joined inviathusis a subregion ofSincecannot be joined intoandis open we deduce
Now use the Maximum Principle. Ifthensince if it were we could enlarge slightly. Thusfor
By applying the Maximum Principle to each polynomial functiononwe obtainforandcontradicting thatsois connected.
]]>Letbe a simply connected region, letbe a simple closed contour inandbe a function analytic onThenfor any pointinside
Proof: Consider the integral
By the shrinking contour theorem we can replaceby any circleof radiusand centrelying insideto obtain(1)
Let
using the parametrization
Then
is continuous atso for eachthere issuch that
Now chooseto be any positive number such thatthen
forin
Hencesinceis the length ofSinceis arbitrarily small
then from (1)
Cauchy's integral formula can be used in a variety of ways, of which more later.
]]>The circulation ofalongiswhereis evaluated along the contour
The flux ofacross
For eachhas magnitude 1 sofor somethenand
so thatand
If for a velocity functionaround every closed curve in the fluid whose motion is described by the velocity field then the velocity field is said to be circulation free.
Example: The flux ofacross the unit circle is
The circulation ofalongis
]]>A functionwhich is a primitive of a complex velocity functionis called a complex potential function for the flow. Taking the complex conjugate of (1) above givesfor
The circulation ofalong a curvedrawn in the fluid iswhere the integral is evaluated alongand the flux ofacrossiswhere the integral is evaluated along
Hencewhere the integral is evaluated along
Provided that %GAMMA lies within the simply connected subregion S on which the complex potential %OMEGA is defined, we may apply the Fundamental Theorem of Algebra to deduce
whereandare the start and end points of the curve
Henceand
Hence the circulation or flux along any contour betweenanddepends only on the points %alpha and %beta and is independent of the contour.
If the flow at each point is tangential to a curvethen the flow acrossat each point ofis zero. Curves with this property are called streamlines and represent snapshots of the motion of the fluid. If a curve is a streamline thenfor all pointsandonso that
]]>A function is conformal if it is conformal on its domain. In fact, any analytic function is conformal at any pointfor which
Proof:higher order terms.
higher order terms.
Subtract these two to givehigher order terms. The tangent vectorsandare both rotated bybut the angle between them is unchanged. In general of course entire regions are mapped. Some examples are shown below. The grid lines in the plane are transformed onto the curves. Since the angle between any tow grid lines is a right angle, so is the angle between any two curves.
]]>for someandfor
then the sequencesatisfiesfor
andandare called conjugate iteration sequences.
In practice the functionis usually found by substitutingandintoand rearranging.
Since the sequenceis the image of the sequenceunder the functionboth sequences must have the same behaviour of convergence and continuity and ifis a fixed point ofthenis a fixed point of
If the conjugating function is to be one to one and entire then it must be of the form
Example: Show that the sequence
is conjugate to the sequencewith conjugating function
Note first thatis one to one onIfthen sobecomes
for
This simplifies to
so thatandare conjugate functions and the sequencesandare conjugate sequences.
]]>Theorem
If a continuous functionoperates on a closed and bounded – compact – setthen the imageis also closed and bounded or compact.
Proof
By the extreme value theorem,is bounded. If we can prove thatis open, then we have proved thatis closed. Since it is also bounded it is compact.
Suppose therefore, thatWe want to find an open disc centred atlying entirely inConsiderwhich is non – zero onsincefor and is continuous on
By the extreme value theorem there existssuch that
for all
that is
for all
whereso the open disc with centreand radiuslies entirely insois compact.
]]>If the radius of convergence is some finite number r, then the power series converges for all z<r and diverges for all z>r, but for z=r, the series may either converge or diverge, or converge at some points absz}=r and diverge for others.
converges for alland diverges for allIfthe series also diverges.
has radius of convergence 1 and converges at every point of the disc of convergenceby comparison with the serieswhich is convergent.
has radius of convergence 1.
If z=1 the series becomes which diverges.
If z=-1 the series becomes which converges so the series converges at every point of the disc of convergenceby comparison with the serieswhich is convergent.
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