Difference between Analytic and Holomorphic function The 2019 Stack Overflow Developer Survey Results Are In Unicorn Meta Zoo #1: Why another podcast? Announcing the arrival of Valued Associate #679: Cesar ManaraAnalytic vs HolomorphicWhat does it mean for a function to be holomorphic?What are some tricks for checking a complex function's analyticity?Taylor serie convergence and holomorphic functionsIs $sum c_n z^n$ analytic when $c_n$ is Banach-valued?On every simply connected domain, there exists a holomorphic function with no analytic continuation.Holomorphic implies analyticCan a complex function be holomorphic only on a proper closed subset of $mathbbC$?history of holomorphic implies analytic and goursat theoremHolomorphic function which is not the derivative of a holomorphic functionA function is analytic if and only if it is holomorphicHolomorphic vs Analytic functions in multiple complex variablesUnderstand analytic function definitionAbout the definition of holomorphic function
Can I visit the Trinity College (Cambridge) library and see some of their rare books
How to determine omitted units in a publication
Windows 10: How to Lock (not sleep) laptop on lid close?
One-dimensional Japanese puzzle
How do I design a circuit to convert a 100 mV and 50 Hz sine wave to a square wave?
Is every episode of "Where are my Pants?" identical?
What to do when moving next to a bird sanctuary with a loosely-domesticated cat?
Make it rain characters
Is there a writing software that you can sort scenes like slides in PowerPoint?
should truth entail possible truth
What was the last x86 CPU that did not have the x87 floating-point unit built in?
Can a flute soloist sit?
What happens to a Warlock's expended Spell Slots when they gain a Level?
Loose spokes after only a few rides
Why did Peik Lin say, "I'm not an animal"?
Why are PDP-7-style microprogrammed instructions out of vogue?
Are spiders unable to hurt humans, especially very small spiders?
Can we generate random numbers using irrational numbers like π and e?
Can the DM override racial traits?
Homework question about an engine pulling a train
Does Parliament need to approve the new Brexit delay to 31 October 2019?
Simulating Exploding Dice
Why not take a picture of a closer black hole?
Can withdrawing asylum be illegal?
Difference between Analytic and Holomorphic function
The 2019 Stack Overflow Developer Survey Results Are In
Unicorn Meta Zoo #1: Why another podcast?
Announcing the arrival of Valued Associate #679: Cesar ManaraAnalytic vs HolomorphicWhat does it mean for a function to be holomorphic?What are some tricks for checking a complex function's analyticity?Taylor serie convergence and holomorphic functionsIs $sum c_n z^n$ analytic when $c_n$ is Banach-valued?On every simply connected domain, there exists a holomorphic function with no analytic continuation.Holomorphic implies analyticCan a complex function be holomorphic only on a proper closed subset of $mathbbC$?history of holomorphic implies analytic and goursat theoremHolomorphic function which is not the derivative of a holomorphic functionA function is analytic if and only if it is holomorphicHolomorphic vs Analytic functions in multiple complex variablesUnderstand analytic function definitionAbout the definition of holomorphic function
$begingroup$
A function $f : mathbbC rightarrow mathbbC$ is said to be holomorphic in an open set $A subset mathbbC$ if it is differentiable at each point of the set $A$.
The function $f : mathbbC rightarrow mathbbC$ is said to be analytic if it has power series representation.
We can prove that the two concepts are same for a single variable complex functions. So why these two different terms? Is there any difference between these two concepts in general, please give example.
Thank you for your help.
complex-analysis definition
edited Sep 16 '18 at 5:20
SRJ
1,8981620
asked Nov 20 '13 at 0:47
DuttaDutta
3,89952443
$endgroup$
add a comment |
$begingroup$
A function $f : mathbbC rightarrow mathbbC$ is said to be holomorphic in an open set $A subset mathbbC$ if it is differentiable at each point of the set $A$.
The function $f : mathbbC rightarrow mathbbC$ is said to be analytic if it has power series representation.
We can prove that the two concepts are same for a single variable complex functions. So why these two different terms? Is there any difference between these two concepts in general, please give example.
Thank you for your help.
complex-analysis definition
edited Sep 16 '18 at 5:20
SRJ
1,8981620
asked Nov 20 '13 at 0:47
DuttaDutta
3,89952443
$endgroup$
12
$begingroup$
Your definitions should be the other way around. You have two concepts each one has its own name, (historically ) it just turn out they define the same functions on the complex numbers.
$endgroup$
– azarel
Nov 20 '13 at 0:53
4
$begingroup$
I've always understood it to be the other way around: "analytic" means it is equal to it power series within some non-zero radius of convergence, and "holomorphic" means it's differentiable.
$endgroup$
– Michael Hardy
Nov 20 '13 at 0:54
1
$begingroup$
As already said, you switched the definitions. The introduction of this link will probably clarify a little more.
$endgroup$
– Mateus Sampaio
Nov 20 '13 at 0:56
add a comment |
$begingroup$
A function $f : mathbbC rightarrow mathbbC$ is said to be holomorphic in an open set $A subset mathbbC$ if it is differentiable at each point of the set $A$.
The function $f : mathbbC rightarrow mathbbC$ is said to be analytic if it has power series representation.
We can prove that the two concepts are same for a single variable complex functions. So why these two different terms? Is there any difference between these two concepts in general, please give example.
Thank you for your help.
complex-analysis definition
edited Sep 16 '18 at 5:20
SRJ
1,8981620
asked Nov 20 '13 at 0:47
DuttaDutta
3,89952443
$endgroup$
A function $f : mathbbC rightarrow mathbbC$ is said to be holomorphic in an open set $A subset mathbbC$ if it is differentiable at each point of the set $A$.
The function $f : mathbbC rightarrow mathbbC$ is said to be analytic if it has power series representation.
We can prove that the two concepts are same for a single variable complex functions. So why these two different terms? Is there any difference between these two concepts in general, please give example.
Thank you for your help.
complex-analysis definition
complex-analysis definition
edited Sep 16 '18 at 5:20
SRJ
1,8981620
asked Nov 20 '13 at 0:47
DuttaDutta
3,89952443
edited Sep 16 '18 at 5:20
SRJ
1,8981620
asked Nov 20 '13 at 0:47
DuttaDutta
3,89952443
edited Sep 16 '18 at 5:20
SRJ
1,8981620
edited Sep 16 '18 at 5:20
SRJ
1,8981620
edited Sep 16 '18 at 5:20
SRJ
1,8981620
1,8981620
asked Nov 20 '13 at 0:47
DuttaDutta
3,89952443
asked Nov 20 '13 at 0:47
DuttaDutta
3,89952443
asked Nov 20 '13 at 0:47
DuttaDutta
3,89952443
3,89952443
12
$begingroup$
Your definitions should be the other way around. You have two concepts each one has its own name, (historically ) it just turn out they define the same functions on the complex numbers.
$endgroup$
– azarel
Nov 20 '13 at 0:53
4
$begingroup$
I've always understood it to be the other way around: "analytic" means it is equal to it power series within some non-zero radius of convergence, and "holomorphic" means it's differentiable.
$endgroup$
– Michael Hardy
Nov 20 '13 at 0:54
1
$begingroup$
As already said, you switched the definitions. The introduction of this link will probably clarify a little more.
$endgroup$
– Mateus Sampaio
Nov 20 '13 at 0:56
add a comment |
12
$begingroup$
Your definitions should be the other way around. You have two concepts each one has its own name, (historically ) it just turn out they define the same functions on the complex numbers.
$endgroup$
– azarel
Nov 20 '13 at 0:53
4
$begingroup$
I've always understood it to be the other way around: "analytic" means it is equal to it power series within some non-zero radius of convergence, and "holomorphic" means it's differentiable.
$endgroup$
– Michael Hardy
Nov 20 '13 at 0:54
1
$begingroup$
As already said, you switched the definitions. The introduction of this link will probably clarify a little more.
$endgroup$
– Mateus Sampaio
Nov 20 '13 at 0:56
12
12
$begingroup$
Your definitions should be the other way around. You have two concepts each one has its own name, (historically ) it just turn out they define the same functions on the complex numbers.
$endgroup$
– azarel
Nov 20 '13 at 0:53
$begingroup$
Your definitions should be the other way around. You have two concepts each one has its own name, (historically ) it just turn out they define the same functions on the complex numbers.
$endgroup$
– azarel
Nov 20 '13 at 0:53
4
4
$begingroup$
I've always understood it to be the other way around: "analytic" means it is equal to it power series within some non-zero radius of convergence, and "holomorphic" means it's differentiable.
$endgroup$
– Michael Hardy
Nov 20 '13 at 0:54
$begingroup$
I've always understood it to be the other way around: "analytic" means it is equal to it power series within some non-zero radius of convergence, and "holomorphic" means it's differentiable.
$endgroup$
– Michael Hardy
Nov 20 '13 at 0:54
1
1
$begingroup$
As already said, you switched the definitions. The introduction of this link will probably clarify a little more.
$endgroup$
– Mateus Sampaio
Nov 20 '13 at 0:56
$begingroup$
As already said, you switched the definitions. The introduction of this link will probably clarify a little more.
$endgroup$
– Mateus Sampaio
Nov 20 '13 at 0:56
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
So why these two different terms?
Because the history of mathematical terms is long and complicated. At least we stopped talking about monogenic functions and regular functions, which are two more terms for the same concept (as far as complex analysis is concerned). Quoting HOMT site:
In modern analysis the term ANALYTIC FUNCTION is used in two ways: (of a complex function) having a complex derivative at every point of its domain, and in consequence possessing derivatives of all orders and agreeing with its Taylor series locally; (of a real function) possessing derivatives of all orders and agreeing with its Taylor series locally.
Since the first usage is so popular (due to the ubiquity of power series in complex analysis, where they exist for every differentiable function), one will often say real-analytic when referring to the usage of the second kind.
Also from HOMT, an explanation of what analytic meant in the less rigorous age of analysis:
[In Lagrange's] Théorie des Fonctions Analytiques (1797) [...] an analytic function simply signified a function of the kind treated in analysis. The connection between the usage of Lagrange and modern usage is explained by Judith V. Grabiner in her The Origins of Cauchy’s Rigorous Calculus: "For Lagrange, all the applications of calculus ... rested on those properties of functions which could be learned by studying their Taylor series developments ... Weierstrass later exploited this idea in his theory of functions of a complex variable, retaining Lagrange’s term "analytic function" to designate, for Weierstrass, a function of a complex variable with a convergent Taylor series."
As for "holomorphic": in complex analysis we often encounter both Taylor series and Laurent series. For the latter, it matters very much whether the number of negative powers is finite or infinite. To enunciate these distinctions, the words holomorphic and meromorphic were introduced. Meromorphic allows poles (i.e., finitely many negative powers in the Laurent series), while holomorphic does not. From a certain viewpoint (the Riemann sphere), meromorphic functions are no worse than holomorphic ones; while at other times, the presence of poles changes the situation.
answered Nov 20 '13 at 13:31
user103402user103402
1,7632933
$endgroup$
2
$begingroup$
and all. Thank you for your information.
$endgroup$
– Dutta
Nov 21 '13 at 0:43
$begingroup$
Lovely explanation.
$endgroup$
– Amey Joshi
Mar 7 '17 at 15:22
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f573984%2fdifference-between-analytic-and-holomorphic-function%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
So why these two different terms?
Because the history of mathematical terms is long and complicated. At least we stopped talking about monogenic functions and regular functions, which are two more terms for the same concept (as far as complex analysis is concerned). Quoting HOMT site:
In modern analysis the term ANALYTIC FUNCTION is used in two ways: (of a complex function) having a complex derivative at every point of its domain, and in consequence possessing derivatives of all orders and agreeing with its Taylor series locally; (of a real function) possessing derivatives of all orders and agreeing with its Taylor series locally.
Since the first usage is so popular (due to the ubiquity of power series in complex analysis, where they exist for every differentiable function), one will often say real-analytic when referring to the usage of the second kind.
Also from HOMT, an explanation of what analytic meant in the less rigorous age of analysis:
[In Lagrange's] Théorie des Fonctions Analytiques (1797) [...] an analytic function simply signified a function of the kind treated in analysis. The connection between the usage of Lagrange and modern usage is explained by Judith V. Grabiner in her The Origins of Cauchy’s Rigorous Calculus: "For Lagrange, all the applications of calculus ... rested on those properties of functions which could be learned by studying their Taylor series developments ... Weierstrass later exploited this idea in his theory of functions of a complex variable, retaining Lagrange’s term "analytic function" to designate, for Weierstrass, a function of a complex variable with a convergent Taylor series."
As for "holomorphic": in complex analysis we often encounter both Taylor series and Laurent series. For the latter, it matters very much whether the number of negative powers is finite or infinite. To enunciate these distinctions, the words holomorphic and meromorphic were introduced. Meromorphic allows poles (i.e., finitely many negative powers in the Laurent series), while holomorphic does not. From a certain viewpoint (the Riemann sphere), meromorphic functions are no worse than holomorphic ones; while at other times, the presence of poles changes the situation.
answered Nov 20 '13 at 13:31
user103402user103402
1,7632933
$endgroup$
2
$begingroup$
and all. Thank you for your information.
$endgroup$
– Dutta
Nov 21 '13 at 0:43
$begingroup$
Lovely explanation.
$endgroup$
– Amey Joshi
Mar 7 '17 at 15:22
add a comment |
$begingroup$
So why these two different terms?
Because the history of mathematical terms is long and complicated. At least we stopped talking about monogenic functions and regular functions, which are two more terms for the same concept (as far as complex analysis is concerned). Quoting HOMT site:
In modern analysis the term ANALYTIC FUNCTION is used in two ways: (of a complex function) having a complex derivative at every point of its domain, and in consequence possessing derivatives of all orders and agreeing with its Taylor series locally; (of a real function) possessing derivatives of all orders and agreeing with its Taylor series locally.
Since the first usage is so popular (due to the ubiquity of power series in complex analysis, where they exist for every differentiable function), one will often say real-analytic when referring to the usage of the second kind.
Also from HOMT, an explanation of what analytic meant in the less rigorous age of analysis:
[In Lagrange's] Théorie des Fonctions Analytiques (1797) [...] an analytic function simply signified a function of the kind treated in analysis. The connection between the usage of Lagrange and modern usage is explained by Judith V. Grabiner in her The Origins of Cauchy’s Rigorous Calculus: "For Lagrange, all the applications of calculus ... rested on those properties of functions which could be learned by studying their Taylor series developments ... Weierstrass later exploited this idea in his theory of functions of a complex variable, retaining Lagrange’s term "analytic function" to designate, for Weierstrass, a function of a complex variable with a convergent Taylor series."
As for "holomorphic": in complex analysis we often encounter both Taylor series and Laurent series. For the latter, it matters very much whether the number of negative powers is finite or infinite. To enunciate these distinctions, the words holomorphic and meromorphic were introduced. Meromorphic allows poles (i.e., finitely many negative powers in the Laurent series), while holomorphic does not. From a certain viewpoint (the Riemann sphere), meromorphic functions are no worse than holomorphic ones; while at other times, the presence of poles changes the situation.
answered Nov 20 '13 at 13:31
user103402user103402
1,7632933
$endgroup$
2
$begingroup$
and all. Thank you for your information.
$endgroup$
– Dutta
Nov 21 '13 at 0:43
$begingroup$
Lovely explanation.
$endgroup$
– Amey Joshi
Mar 7 '17 at 15:22
add a comment |
$begingroup$
So why these two different terms?
Because the history of mathematical terms is long and complicated. At least we stopped talking about monogenic functions and regular functions, which are two more terms for the same concept (as far as complex analysis is concerned). Quoting HOMT site:
In modern analysis the term ANALYTIC FUNCTION is used in two ways: (of a complex function) having a complex derivative at every point of its domain, and in consequence possessing derivatives of all orders and agreeing with its Taylor series locally; (of a real function) possessing derivatives of all orders and agreeing with its Taylor series locally.
Since the first usage is so popular (due to the ubiquity of power series in complex analysis, where they exist for every differentiable function), one will often say real-analytic when referring to the usage of the second kind.
Also from HOMT, an explanation of what analytic meant in the less rigorous age of analysis:
[In Lagrange's] Théorie des Fonctions Analytiques (1797) [...] an analytic function simply signified a function of the kind treated in analysis. The connection between the usage of Lagrange and modern usage is explained by Judith V. Grabiner in her The Origins of Cauchy’s Rigorous Calculus: "For Lagrange, all the applications of calculus ... rested on those properties of functions which could be learned by studying their Taylor series developments ... Weierstrass later exploited this idea in his theory of functions of a complex variable, retaining Lagrange’s term "analytic function" to designate, for Weierstrass, a function of a complex variable with a convergent Taylor series."
As for "holomorphic": in complex analysis we often encounter both Taylor series and Laurent series. For the latter, it matters very much whether the number of negative powers is finite or infinite. To enunciate these distinctions, the words holomorphic and meromorphic were introduced. Meromorphic allows poles (i.e., finitely many negative powers in the Laurent series), while holomorphic does not. From a certain viewpoint (the Riemann sphere), meromorphic functions are no worse than holomorphic ones; while at other times, the presence of poles changes the situation.
answered Nov 20 '13 at 13:31
user103402user103402
1,7632933
$endgroup$
So why these two different terms?
Because the history of mathematical terms is long and complicated. At least we stopped talking about monogenic functions and regular functions, which are two more terms for the same concept (as far as complex analysis is concerned). Quoting HOMT site:
In modern analysis the term ANALYTIC FUNCTION is used in two ways: (of a complex function) having a complex derivative at every point of its domain, and in consequence possessing derivatives of all orders and agreeing with its Taylor series locally; (of a real function) possessing derivatives of all orders and agreeing with its Taylor series locally.
Since the first usage is so popular (due to the ubiquity of power series in complex analysis, where they exist for every differentiable function), one will often say real-analytic when referring to the usage of the second kind.
Also from HOMT, an explanation of what analytic meant in the less rigorous age of analysis:
[In Lagrange's] Théorie des Fonctions Analytiques (1797) [...] an analytic function simply signified a function of the kind treated in analysis. The connection between the usage of Lagrange and modern usage is explained by Judith V. Grabiner in her The Origins of Cauchy’s Rigorous Calculus: "For Lagrange, all the applications of calculus ... rested on those properties of functions which could be learned by studying their Taylor series developments ... Weierstrass later exploited this idea in his theory of functions of a complex variable, retaining Lagrange’s term "analytic function" to designate, for Weierstrass, a function of a complex variable with a convergent Taylor series."
As for "holomorphic": in complex analysis we often encounter both Taylor series and Laurent series. For the latter, it matters very much whether the number of negative powers is finite or infinite. To enunciate these distinctions, the words holomorphic and meromorphic were introduced. Meromorphic allows poles (i.e., finitely many negative powers in the Laurent series), while holomorphic does not. From a certain viewpoint (the Riemann sphere), meromorphic functions are no worse than holomorphic ones; while at other times, the presence of poles changes the situation.
answered Nov 20 '13 at 13:31
user103402user103402
1,7632933
edited Nov 20 '13 at 13:43
answered Nov 20 '13 at 13:31
user103402user103402
1,7632933
answered Nov 20 '13 at 13:31
user103402user103402
1,7632933
answered Nov 20 '13 at 13:31
user103402user103402
1,7632933
1,7632933
2
$begingroup$
and all. Thank you for your information.
$endgroup$
– Dutta
Nov 21 '13 at 0:43
$begingroup$
Lovely explanation.
$endgroup$
– Amey Joshi
Mar 7 '17 at 15:22
add a comment |
2
$begingroup$
and all. Thank you for your information.
$endgroup$
– Dutta
Nov 21 '13 at 0:43
$begingroup$
Lovely explanation.
$endgroup$
– Amey Joshi
Mar 7 '17 at 15:22
2
2
$begingroup$
and all. Thank you for your information.
$endgroup$
– Dutta
Nov 21 '13 at 0:43
$begingroup$
and all. Thank you for your information.
$endgroup$
– Dutta
Nov 21 '13 at 0:43
$begingroup$
Lovely explanation.
$endgroup$
– Amey Joshi
Mar 7 '17 at 15:22
$begingroup$
Lovely explanation.
$endgroup$
– Amey Joshi
Mar 7 '17 at 15:22
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f573984%2fdifference-between-analytic-and-holomorphic-function%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
12
$begingroup$
Your definitions should be the other way around. You have two concepts each one has its own name, (historically ) it just turn out they define the same functions on the complex numbers.
$endgroup$
– azarel
Nov 20 '13 at 0:53
4
$begingroup$
I've always understood it to be the other way around: "analytic" means it is equal to it power series within some non-zero radius of convergence, and "holomorphic" means it's differentiable.
$endgroup$
– Michael Hardy
Nov 20 '13 at 0:54
1
$begingroup$
As already said, you switched the definitions. The introduction of this link will probably clarify a little more.
$endgroup$
– Mateus Sampaio
Nov 20 '13 at 0:56