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Purim Katan: 
Keil Erech Apayim: If the 14th or 15th of Adar Rishon falls on Monday or Thursday, the prayer of Keil Erech Apayim is omitted on prior to the Torah reading,
Tzidkascha Tzedek: The prayer of Tzidkascha Tzedek is omitted on Shabbos, after Mincha, if the [13th or] 14th or 15th of Adar Rishon coincides with Shabbos.
Hesped: It is forbidden to give a eulogy on the 14th and the 15th of Adar Rishon.
Fast: It is forbidden to fast on the 14th and the 15th of Adar Rishon.
Aveilus: One who is in the midst of sitting Shiva, continues to keep all the mourning customs throughout Purim Katan.
Melacha: There is no prohibition against doing Melacha on Purim Katan.
Megillah reading/Mishloach Manos/Matanos Laevyonim: These Mitzvos are not applicable on Purim Katan.
Festivities: One is to increase in festive meals on the 14th of Adar Rishon. [This applies even in Jerusalem. One is to invite guests to the meal. The meal is to include singing and praise to G-d for the miracle that occurred.] On the 15th there is no obligation to increase in festivities even for those cities which celebrate Purim on the 15th, such as Jerusalem. Nonetheless one is to also increase in joy on the 15th, as concludes the Rama “Tov Leiv Mishteh Tamid”.
Are a Chasan and Kallah to fast on their wedding day if it falls on Purim Katan?
Is the special meal to be eaten on the night of the 14th or the day?
Seemingly the meal is to be eaten during the day of the 14th, as is the law regarding the Purim feast.
May one visit a cemetery Purim Katan?
Yes. One may visit the grave of a relative, such as for the end of Shiva; Shloshim; or Yartzite. However eulogies may not be given.
May a Matzeiva [tombstone] be established on Purim Katan?
This may not be done if eulogies will take place during the occasion.
What is the law if one accidently recited Al Hanissim on Purim Katan?
He is not required to repeat Shemoneh Esrei.
What is the law if one accidently read the Megillah in the first Adar?
One must repeat the reading in Adar II.
 697/1; Last chapter of Shulchan Aruch Orach Chaim
Source of the name Purim Katan: Tzaruch Iyun as to the source for the name Purim Katan in Rishonim and onwards. Perhaps it is taken from the term Pesach Sheiyni which is also called “the small Pesach” or “Pesach Zeira”. However no explicit source has yet to be found to call it this name. [Rebbe in Shaar Hamoadim 20] To note that Admur in the Siddur uses the term “Purim Katan”.
The mention of the 14th and 15th of Adar Rishon is first made in the Mishneh Megila 6b. The Mishneh states that all the laws that apply in the second Adar apply likewise on the first Adar, with exception to reading the Megila and Matanos Laevyonim. The Gemara then brings three opinions regarding this matter. Some [Tana Kama] hold that all the Mitzvos are to be done by Rishon and Sheiyni, with exception to reading the Megila which is only done in Sheiyni. Others rule that all the Mitzvos are done in Sheiyni and nothing is done in Rishon. Others [Rabbi Shimon Ben Gamliel] rule that although there aren’t any Mitzvos done in Rishon, nevertheless it is forbidden to fast and say eulogies. Practically we rule like this latter opinion. [Rebbe Yochanan in Gemara ibid; and so rules Rif and other Rishonim] Nevertheless there is a dispute amongst Rishonim if this prohibition of fasting and eulogies applies in all situations, or only in the case of the Mishneh, that the community mistakenly read the Megillah on Adar Rishon. [See Tur for dispute between Rosh; Rif and other Rishonim] Practically we rule stringently. [Rama ibid]
 Michaber ibid; Siddur Admur
 The reason: As these days are considered Purim Katan. Admur ibid mentions “Purim Katan which is on the 14th 15th”. The reason why Admur mentions both the term Purim Katan and the date of the 14th and 15th is because all private commemorations of miracles are called Purim Katan, and thus Admur adds the dates in order so it be understood that Tachanun is only omitted on the Purim Katan of the 14th and 15th. The reason why Admur also records the term Purim Katan and does not suffice with writing the dates alone is because this teaches us the reason for why Tachanun is omitted, as it is considered Purim Katan. [Rebbe in Shaareiy Hamoadim 22]
 Siddur Admur; Shulchan Gavoa 697/3; Kaf Hachaim 697/3
 Michaber ibid; M”B 696/2
 This applies according to all Nuschaos, even those which pray Nusach Ashkenaz, as is evident from Michaber ibid.
 Siddur Admur; Darkei Moshe 697/2; Levush; P”M 697 A”A 1; M”B 697/2
 Darkei Moshe 697/2; Levush; P”M 697 A”A 1; M”B 697/2
 Michaber ibid; Second opinion in Tur
Other Opinions: Some opinions [First opinion in Tur] rule that even fasting and eulogies are permitted on Purim Katan. [Brought in Michaber ibid] However the custom is like the opinion mentioned above that it is forbidden to say a eulogy or fast. [Rama ibid]
 Michaber ibid; Second opinion in Tur
 Elya Raba 697/1; P”M 697 M”Z 1; M”B 697/3
 Beis Yosef; Ateres Zekeinim; Elya Raba 697/1; P”M 697 A”A 1
 Michaber ibid
 M”A 697/1; Elya Raba 697/1; M”B 697/1
 The reason: As the Megillah is not being read. [ibid]
 Rama ibid “Some Opinions [Ran] rule one must increase in merry making and joy on this day. This however is not the custom. Practically one is to nevertheless increase slightly in a festive meal in order to fulfill one’s obligation according to those that are stringent.”
 The Rama rules that one is to increase slightly in a festive meal and then concludes with the verse “Vetov Leiv Mishte Tamid”. The Taz 697/2 learns this to mean that the Rama in truth is hinting in this conclusion that one is to have a great feast [and not just slightly] as did Rav Yechiel Miparish and others, similar to the actual feast of Purim in Adar Sheyni. [Taz ibid; M”B 697/5] The reason that the Rama does not explicitly write this ruling is because obligating one to have joy itself shows that the joy does not come on its own right and must hence be commanded. For this reason the Rama used a term which emphasizes that this joy is so high that it has to come on its own without obligation. [Rebbe in Likkutei Sichos 26]
 Rama 697/1
The reason: In order so one fulfill his obligation according to those opinions which require one to increase in merry making and joy on this day. Thus the Rama concludes that despite that this is not the custom one is to nevertheless increase slightly.
 P”M 697 A”A 1; M”B 697/4
 Taz 697/1 in name of Rav Yechiel Miparish, brought in Tashbatz 178; M”B 697/5
 See Rama 670/2 regarding festive meals on Chanukah, of which there is also a dispute, “one is to sing praise and song during the meals and in this way the meal is considered a Seudas Mitzvah [according to all opinions]”
 M”B 697/4, as is implied from Rama which mentions specifically the 14th.
Other Poskim: See Minchas Yitzchak 10/58 that the Smak holds one is to increase in merry making on both days. This especially applies to residents of Jerusalem. [See Piskeiy Teshuvos 697/2]
 Elya Raba 697/2 says that perhaps merry making on the 15th is also included in the concluding statement of the Rama, to always be merry. The Rebbe concludes that one is to be stringent, as the Rama himself concludes “Tov Leiv Mishteh Tamid”, that whenever there is an opinion which requires one to be happy one is to do so. [Shaareiy Hamoadim Adar 23]
 Arugas Bosem 182; Nitei Gavriel 16/4; Nemukei Orach Chaim 697
The reason: As it is forbidden from the letter of the law to fast on Purim Katan, hence how can we allow a Chasan to fast? This is similar to the law forbidding a Chasan to fast on Issru Chag. [429/17]
 Oatzer Minhagei Chabad 42; Toras Menachem 5711 p. 252; This directive was given to Rav Tzevi Hirsh Chitrig on the occasion of his wedding that took place on Purim Katan 1951. Rav Eizinshtat asked the Rebbe afterwards why he gave such a directive and the Rebbe answered “As I was asked so I answered. I was asked “Is it worthwhile to fast” and I replied that it is. [Mikadeish Yisrael p. 91]
The reason: As perhaps not fasting on Purim Katan is only a custom and not forbidden from the letter of the law. [see Hiskashrus 1020 footnote 6]
 Shraga Hameir 7/152; Nitei Gavriel Purim Teshuvah 2
 Nimukei Orach Chaim ibid; Piskeiy Teshuvos 697/1 footnote 1
 Piskeiy Teshuvos 697/1; See Tzitz Eliezer 18/43
 Nitei Gavriel 16/3; Piskeiy Teshuvos 697/1
 M”B 697/1 in name of Chemed Moshe; Kaf Hachaim 697/2; See Derech Hachaim 33/29, 46, 52, 72 and Ketzos Hashulchan 21/4 that this applies anytime of the year, in which if one accidently said Al Hanissim he is not required to repeat Shemoneh Esrei being that he did not say an absolute lie but simply added to his prayer. So is also implied from Admur 108/18 which does not mention Al Hanissim in the dispute mentioned regarding one who accidently said Yaleh Veyavo in Shemoneh Esrei. [Ketzos Hashulchan ibid footnote 8]
 Darkei Moshe 688/7; M”A 688/11; M”B 688/23; Kaf Hachaim 688/60