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# Section 2.4 Limits and Continuity

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C02-4.mws

COMPLEX ANALYSIS: Maple Worksheets,  2001
(c) John H. Mathews          Russell W. Howell

mathews@fullerton.edu     howell@westmont.edu

Complimentary software to accompany the textbook:

COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
Jones and Bartlett Publishers, Inc.,      40  Tall  Pine  Drive,      Sudbury,  MA  01776

Tele.  (800) 832-0034;      FAX:  (508)  443-8000,      E-mail:  mkt@jbpub.com,      http://www.jbpub.com/

CHAPTER 2   COMPLEX FUNCTIONS

Section 2.4  Limits and Continuity

Let be a real-valued function of the two real variables and .  Recall that has the as ( ) approaches ( ) provided that the value of can be made to get as close as we please to the value by taking ( ) to be sufficiently close to ( ).

Example 2.14, Page 69.
The function has the limit as ( , ) approaches ( , ) .

 > t:='t': u:='u': x:='x': y:='y': u := proc(x,y)  2*x^3/(x^2+y^2)  end: `u(x,y) ` = u(x,y); ` `; lim1 := limit(u(x,y), x=0): lim2 := limit(lim1, y=0): `limit  u(x,y)  as  x->0 ` = lim1; `and`; `limit  u(x,y)  as  x->0 and y->0 ` = lim2;     > `u(x,y) ` = u(x,y); ` `; lim1 := limit(u(x,y), y=0): lim2 := limit(lim1, x=0): `limit  u(x,y)  as  y->0 ` = lim1; `and`; `limit  u(x,y)  as  y->0 and x->0 ` = lim2;     > U := subs({x=r*cos(t),y=r*sin(t)},u(x,y)): `u(r cos t,r sin t) ` = U; ` `; lim1 := limit(U, r=0): `limit  u(r cos t,r sin t)  as  r->0 ` = lim1;   So, along all lines through the origin, the limit is .

Example 2.15, Page 70.
The function does
NOT have a limit as  ( , )  approaches  ( , ) .

 > t:='t': u:='u': x:='x': y:='y': u := proc(x,y)  x*y/(x^2+y^2)  end: `u(x,y) ` = u(x,y); ` `; lim1 := limit(u(x,y), x=0): lim2 := limit(lim1, y=0): `limit  u(x,y)  as  x->0 ` = lim1; `and`; `limit  u(x,y)  as  x->0 and y->0 ` = lim2;     > `u(x,y) ` = u(x,y); ` `; lim1 := limit(u(x,y), y=0): lim2 := limit(lim1, x=0): `limit  u(x,y)  as  y->0 ` = lim1; `and`; `limit  u(x,y)  as  y->0 and x->0 ` = lim2;     > U := subs({x=r*cos(t),y=r*sin(t)},u(x,y)): `u(r cos t,r sin t) ` = U; ` `; lim1 := limit(U, r=0): `limit  u(r cos t,r sin t)  as  r->0 ` = simplify(lim1);   Since this value is dependent on the angle of approach to , does
NOT have a limit  as  ( , )  approaches  ( , ) .

Theorem 2.1   Let be a complex function that is defined in some neighborhood of ,

except perhaps at .  Then = if and only if and .

Limits of complex functions are formally the same as in the case of real functions, and the sum, difference, product, and quotient of functions have limits given by the sum, difference, product, and quotient of the respective limits. These proofs are left as exercises.

Example 2.17, Page 73.
Find for .

 > f:='f': z:='z': f := z -> z^2 - 2*z + 1: `f(z) ` = f(z); ` `; `limit  f(z)  as  z->1+i ` = limit(f(z), z=1+I); `Also, the value of f(1+i) is:`; `f(1+i) ` = f(1+I);     Example 2.18, Page 75.
Show that the polynomial function given by + ... + is continuous at each point in the complex plane.
For illustration,
we use .

 > P:='P': z:='z': z0:='z0': P := z -> sum('a[k]'*z^k, 'k'=0..5): `P(z) ` = P(z); lim := limit(P(z), z=z0): `limit P(z)  as  z->z0 ` = lim; `Also, the value of P(z0) is:`; `P(z0) ` = P(z0); ` `; `P(z0) = limit P(z)  as  z->z0 `; evalb(limit(P(z), z=z0) = P(z0));       Example 2.19, Page 76.  Find for .

 > f:='f': z:='z': f := z -> (z^2 - 2*I)/(z^2 - 2*z + 2): `f(z) ` = f(z); fun := f(1+I): `f(1+I) ` = undefined; `However,`; lim := limit(f(z) ,z=1+I): `limit  f(z)  as  z->1+i ` = lim; Error, (in f) numeric exception: division by zero   > f:='f': F:='F': z:='z': Z :='Z': f := z -> (z^2 - 2*I)/(z^2 - 2*z + 2): fact := factor(f(Z)): F := z -> subs(Z=z,fact): `f(z) ` = f(z); `Simplify the function.`; `F(z) ` = F(z); ` `; `Evaluate F(z) at  z = 1+i`; `F(1+i) ` = F(1+I);      >

End of Section 2.4.