84:
74:
53:
22:
165:
a) It is defined as the limit at a of the function existing (i.e. the limit is the same as you approach from all sides, but the function is defined as something else when it is actually at a. (sorry I don't feel like inserting the latex right now).
215:
The proof given here strikes me as rather strange. In particular, it's necessary to show that the Taylor series won't have a first order term. And what is the motivation for multiplying by (z-a)^2 rather than by (z-a)? Ahlfors,
192:
Yes, this phenomenon is unique to complex analysis. If the function is bounded at the singularity, that's enough to guarantee not only its continuity, but even infinite differentiability. I clarified this in the article.
140:
162:
I believe that your statement of
Reimann's theorem is incorrect. While I can't seem to find a good description of the theorem online anywhere I am pretty sure that:
170:=1 we have a function that is obviously bounded at 0, but this is not a removable singularity. No redefinition of the Heaviside function will make this continuous.
318:
130:
313:
106:
173:
Perhaps this definition of
Reimann's theorem makes more sense in the complex analysis sense, but if so you should be more specific. --anon
97:
58:
254:) exists (namely, it is 0), but this should be stated explicitly in the proof. The first two terms of the Taylor series of
33:
198:
21:
39:
83:
220:(on page 124 in the 3rd edition, 1979) gives a neat proof using the Cauchy integral formula instead.
194:
178:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
225:
89:
73:
52:
185:
295:
235:
I agree that the proof is not very clear. I guess that the motivation for multiplying by (
307:
221:
182:
298:
229:
202:
187:
169:
b) In the example of the
Heaviside functin where f(x)=0 x<0 and f(x)=1 x: -->
102:
292:
79:
15:
181:. Perhaps a different wording could make this clearer. —
101:, a collaborative effort to improve the coverage of
8:
177:I believe Riemann's theorem applies only to
19:
47:
49:
7:
95:This article is within the scope of
38:It is of interest to the following
14:
319:Low-priority mathematics articles
115:Knowledge:WikiProject Mathematics
314:Start-Class mathematics articles
118:Template:WikiProject Mathematics
82:
72:
51:
20:
135:This article has been rated as
1:
109:and see a list of open tasks.
299:11:40, 25 January 2010 (UTC)
230:05:49, 25 January 2010 (UTC)
203:01:02, 9 December 2005 (UTC)
188:22:07, 8 December 2005 (UTC)
335:
134:
67:
46:
141:project's priority scale
98:WikiProject Mathematics
28:This article is rated
179:holomorphic functions
243:) is to ensure that
158:Incorrect statement?
121:mathematics articles
258:then vanish since
90:Mathematics portal
34:content assessment
155:
154:
151:
150:
147:
146:
326:
285:
248:
218:Complex Analysis
211:Method of proof?
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
334:
333:
329:
328:
327:
325:
324:
323:
304:
303:
283:
279:
264:
246:
213:
195:Oleg Alexandrov
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
332:
330:
322:
321:
316:
306:
305:
302:
301:
277:
262:
212:
209:
208:
207:
206:
205:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
331:
320:
317:
315:
312:
311:
309:
300:
297:
294:
290:
286:
276:
272:
268:
261:
257:
253:
249:
242:
238:
234:
233:
232:
231:
227:
223:
219:
210:
204:
200:
196:
191:
190:
189:
186:
184:
180:
176:
175:
174:
171:
167:
163:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
288:
281:
274:
270:
266:
259:
255:
251:
244:
240:
236:
217:
214:
172:
168:
164:
161:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
308:Categories
273:) = 0 and
291:) = 0. —
222:Jowa fan
183:Caesura
139:on the
36:scale.
293:Emil
226:talk
199:talk
131:Low
310::
296:J.
280:=
265:=
239:−
228:)
201:)
289:a
287:(
284:'
282:h
278:1
275:a
271:a
269:(
267:h
263:0
260:a
256:h
252:a
250:(
247:'
245:h
241:a
237:z
224:(
197:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
