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A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example of this is the apparent singularity at the 90 degree latitude in spherical coordinates. An object moving due north (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (in the case of the example, jumping from longitude 0 to longitude 180 degrees). This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity (e.g., by replacing the latitude/longitude representation with an -vector representation).
A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example of this is the apparent singularity at the 90 degree latitude in spherical coordinates. An object moving due north (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (in the case of the example, jumping from longitude 0 to longitude 180 degrees). This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity (e.g., by replacing the latitude/longitude representation with an -vector representation).
当在一个坐标系中出现明显的奇异性或不连续性时，就会出现坐标奇点，可以通过选择不同的坐标系来消除。这方面的一个例子是在球面坐标系中90度纬度处的明显奇异性。在球体表面正北方移动的物体（例如，沿经度为0度的直线）将突然在极点处经历经度的瞬时变化（在本例中，从经度0跳到经度180度）。然而，这种不连续性只是显而易见的；它是所选坐标系的一个伪影，在极点处是奇异的。不同的坐标系将消除明显的不连续性（例如，用矢量表示代替经纬度表示法）。
如果存在定义在“U”上的全纯函数“g”，且“g”（“a”）非零，且存在一个自然数“n”，使得对所有“z”属于“U”\{“a”},“f”（“z”）=“g”（“z”）/ (“z” – “a”)n，则点“a”为[[极点（复分析）|极]]或“f”的<font color=“#ff8000”>非本质奇点 non-essential singularity</font>。最小的这个数“n”称为“极序”。 <font color=“#ff8000”>非本质奇点</font>处的导数本身也有一个<font color=“#ff8000”>非本质奇点</font>，当“n”增加1时（除非“n”为0，因此<font color="#ff8000">奇点</font>可移除）。
如果存在定义在“U”上的全纯函数“g”，且“g”（“a”）非零，且存在一个自然数“n”，使得对所有“z”属于“U”\{“a”},“f”（“z”）=“g”（“z”）/ (“z” – “a”)n，则点“a”为[[极点（复分析）|极]]或“f”的<font color=“#ff8000”>非本质奇点 non-essential singularity</font>。最小的这个数“n”称为“极序”。 <font color=“#ff8000”>非本质奇点</font>处的导数本身也有一个<font color=“#ff8000”>非本质奇点</font>，当“n”增加1时（除非“n”为0，因此<font color="#ff8000">奇点</font>可移除）。
* The point ''a'' is an [[essential singularity]] of ''f'' if it is neither a removable singularity nor a pole. The point ''a'' is an essential singularity [[iff|if and only if]] the [[Laurent series]] has infinitely many powers of negative degree.<ref name=":1" />
* The point ''a'' is an [[essential singularity]] of ''f'' if it is neither a removable singularity nor a pole. The point ''a'' is an essential singularity [[iff|if and only if]] the [[Laurent series]] has infinitely many powers of negative degree.<ref name=":1" />
如果点“a”既不是可去奇点，也不是极点，则它是“f”的 <font color=“#ff8000”>非本质奇点</font>。点“a”是 <font color=“#ff8000”>非本质奇点</font>[[iff |当且仅当][[Laurent级数]]具有无穷多个负次幂。
如果点“a”既不是可去奇点，也不是极点，则它是“f”的 非本质奇点。点“a”是非本质奇点[[iff |当且仅当][[Laurent级数]]具有无穷多个负次幂。