Application of ionic‐liquid‐supported magnetic dispersive solid‐phase microextraction for the determination of acaricides in fruit juice samples

In this study, ionic liquid (IL) supported magnetic dispersive solid‐phase microextraction was developed and a systematic investigation was conducted on imidazolium ILs for their extraction performance. This nano‐based pretreatment procedure was then applied for the determination of acaricides in fruit juice samples for the first time. A feature of this technique is that the commonly laborious chemical modification of magnetic nanoparticles (MNPs) was skillfully circumvented. Because of the combination of ILs, dispersive liquid–liquid microextraction, and dispersive MNP solid‐phase microextraction, the extraction efficiency can be significantly improved using commercial MNPs. Parameters of the extraction method were investigated by one‐factor‐at‐a‐time approach. The optimal experimental conditions were as follows: emulsification for 2 min by sonication with the addition of 50 μL [C₆MIM][NTf₂] in the dispersive liquid–liquid microextraction step and vortexing for 90 s after adding 40 mg spherical barium ferrite nanoparticles (20 nm). The desorption time was 2 min. Good linearity (0.5–500 ng/mL) and detection limits within the range of 0.05–0.53 ng/mL were achieved. The application of the proposed method was demonstrated by the analysis of real fruit juice samples, in which recoveries between 85.1 and 99.6% were obtained.     

Solid‐phase extraction assisted dispersive liquid–liquid microextraction based on solidification of floating organic droplet to determine sildenafil and its analogues in dietary supplements

A novel analytical method for the simultaneous determination of the concentration of sildenafil and its five analogues in dietary supplements using solid‐phase extraction assisted reversed‐phase dispersive liquid–liquid microextraction based on solidification of floating organic droplet combined with ion‐pairing liquid chromatography with an ultraviolet detector was developed. Parameters that affect extraction efficiency were systematically investigated, including the type of solid‐phase extraction cartridge, pH of the extraction environment, and the type and volume of extraction and dispersive solvent. The method linearity was in the range of 5.0–100 ng/mL for sildenafil, homosildenafil, udenafil, benzylsildenafil, and thiosildenafil and 10–100 ng/mL for acetildenafil. The coefficients of determination were ≥0.996 for all regression curves. The sensitivity values expressed as limit of detection were between 2.5 and 7.5 ng/mL. Furthermore, intraday and interday precisions expressed as relative standard deviations were less than 5.7 and 9.9%, respectively. The proposed method was successfully applied to the analysis of sildenafil and its five analogues in complex dietary supplements.     

Matrix solid‐phase dispersion with chitosan‐zinc oxide nanoparticles combined with flotation‐assisted dispersive liquid–liquid microextraction for the determination of 13 n‐alkanes in soil samples

In this study, chitosan‐zinc oxide nanoparticles were used as a sorbent of miniaturized matrix solid‐phase dispersion combined with flotation‐assisted dispersive liquid–liquid microextraction for the simultaneous determination of 13 n‐alkanes such as C₈H₁₈ and C₂₀H₄₂ in soil samples. The solid samples were directly blended with the chitosan nanoparticles in the solid‐phase dispersion method. The eluent of solid‐phase dispersion was applied as the dispersive solvent for the following flotation‐assisted dispersive liquid–liquid microextraction for further purification and enrichment of the target compounds prior to gas chromatography with flame ionization detection. Under the optimum conditions, good linearity with correlation coefficients in the range 0.9991 < r² < 0.9995 and low detection limits between 0.08 to 2.5 ng/g were achieved. The presented procedure combined the advantages of chitosan‐zinc oxide nanoparticles, solid‐phase dispersion and flotation‐assisted dispersive liquid–liquid microextraction, and could be applied for the determination of n‐alkanes in complicated soil samples with acceptable recoveries.     

Magnetic solid‐phase extraction or dispersive liquid–liquid microextraction for pyrethroid determination in environmental samples

The determination of 15 pyrethroids in soil and water samples was carried out by gas chromatography with mass spectrometry. Compounds were extracted from the soil samples (4 g) using solid–liquid extraction and then salting‐out assisted liquid–liquid extraction. The acetonitrile phase obtained (0.8 mL) was used as a dispersant solvent, to which 75 μL of chloroform was added as an extractant solvent, submitting the mixture to dispersive liquid–liquid microextraction. For the analysis of water samples (40 mL), magnetic solid‐phase extraction was performed using nanocomposites of magnetic nanoparticles and multiwalled carbon nanotubes as sorbent material (10 mg). The mixture was shaken for 45 min at room temperature before separation with a magnet and desorption with 3 mL of acetone using ultrasounds for 5 min. The solvent was evaporated and reconstituted with 100 μL acetonitrile before injection. Matrix‐matched calibration is recommended for quantification of soil samples, while water samples can be quantified by standards calibration. The limits of detection were in the range of 0.03–0.5 ng/g (soil) and 0.09–0.24 ng/mL (water), depending on the analyte. The analyzed environmental samples did not contain the studied pyrethroids, at least above the corresponding limits of detection.     

Sensitive determination of methadone in human serum and urine by dispersive liquid–liquid microextraction based on the solidification of a floating organic droplet followed by HPLC–UV

Dispersive liquid–liquid microextraction based on solidification of floating organic droplet was developed for the extraction of methadone and determination by high‐performance liquid chromatography with UV detection. In this method, no microsyringe or fiber is required to support the organic microdrop due to the usage of an organic solvent with a low density and appropriate melting point. Furthermore, the extractant droplet can be collected easily by solidifying it at low temperature. 1‐Undecanol and methanol were chosen as extraction and disperser solvents, respectively. Parameters that influence extraction efficiency, i.e. volumes of extracting and dispersing solvents, pH, and salt effect, were optimized by using response surface methodology. Under optimal conditions, enrichment factor for methadone was 134 and 160 in serum and urine samples, respectively. The limit of detection was 3.34 ng/mmL in serum and 1.67 ng/mL in urine samples. Compared with the traditional dispersive liquid–liquid microextraction, the proposed method obtained lower limit of detection. Moreover, the solidification of floating organic solvent facilitated the phase transfer. And most importantly, it avoided using high‐density and toxic solvents of traditional dispersive liquid–liquid microextraction method. The proposed method was successfully applied to the determination of methadone in serum and urine samples of an addicted individual under methadone therapy.     

Application of liquid–liquid microextraction for the effective separation and simultaneous determination of 11 pharmaceuticals in wastewater samples using high‐performance liquid chromatography with tandem mass spectrometry

A dispersive liquid–liquid microextraction method for the simultaneous determination of 11 pharmaceuticals has been developed. The method is based on a microextraction procedure applied to wastewater samples from different regions of Hungary followed by high‐performance liquid chromatography with mass spectrometry. The effect of the nature of the extractant, dispersive solvent, different additives, and extraction time were examined on the extraction efficiently of the dispersive liquid–liquid microextraction method. Under optimal conditions, the linearity for determining the pharmaceuticals was in the range of 1–500 ng/mL, with the correlation coefficients ranging from 0.9922 to 0.9995. The limits of detection and limits of quantification were in the range of 0.31–6.65 and 0.93–22.18 ng/mL, respectively.     

Quantitative determination of trace phenazopyridine in human urine samples by hyphenation of dispersive solid‐phase extraction and liquid‐phase microextraction followed by gas chromatography/mass spectrometry analysis

Magnetic dispersive solid‐phase extraction followed by dispersive liquid?liquid microextraction coupled with gas chromatography/mass spectrometry was applied for the quantitative analysis of phenazopyridine in urinary samples. Magnetic dispersive solid‐phase extraction was carried out using magnetic graphene oxide nanoparticles modified by poly(thiophene‐pyrrole) copolymer. The eluting solvent of this step was used as the disperser solvent for the dispersive liquid?liquid microextraction procedure. To reach the maximum efficiency of the method, effective parameters including sorbent amount, adsorption time, type and volume of disperser and extraction solvents, pH of the sample solution, and ionic strength as well as desorption time, and approach were optimized, separately. Characterization of the synthesized sorbent was studied by utilizing infrared spectroscopy, scanning electron microscopy, and energy‐dispersive X‐ray analysis. Calibration curve was linear in the range of 0.5?250 ng/mL (R² = 0.9988) with limits of detection and quantification of 0.1 and 0.5 ng/mL, respectively. Intra‐ and interday precisions (RSD%, n = 3) of the method were in the range of 4.6?5.4% and 4.0?5.5%, respectively, at three different concentration levels. Under the optimal condition, this method was successfully applied for the determination of phenazopyridine in human urine samples. The relative recoveries were obtained in the range of 85.0?89.0%.     

Trace determination of five triazole fungicide residues in traditional Chinese medicine samples by dispersive solid‐phase extraction combined with ultrasound‐assisted dispersive liquid–liquid microextraction and UHPLC–MS/MS

A novel and reliable method for determination of five triazole fungicide residues (triadimenol, tebuconazole, diniconazole, flutriafol, and hexaconazol) in traditional Chinese medicine samples was developed using dispersive solid‐phase extraction combined with ultrasound‐assisted dispersive liquid–liquid microextraction before ultra‐high performance liquid chromatography with tandem mass spectrometry. The clean up of the extract was conducted using dispersive solid‐phase extraction by directly adding sorbents into the extraction solution, followed by shaking and centrifugation. After that, a mixture of 400 μL trichloromethane (extraction solvent) and 0.5 mL of the above supernatant was injected rapidly into water for the dispersive liquid–liquid microextraction procedure. The factors affecting the extraction efficiency were optimized. Under the optimum conditions, the calibration curves showed good linearity in the range of 2.0–400 (tebuconazole, diniconazole, and hexaconazole) and 4.0–800 ng/g (triadimenol and flutriafol) with the regression coefficients higher than 0.9958. The limit of detection and limit of quantification for the present method were 0.5–1.1 and 1.8–4.0 ng/g, respectively. The recoveries of the target analytes ranged from 80.2 to 103.2%. The proposed method has been successfully applied to the analysis of five triazole fungicides in traditional Chinese medicine samples, and satisfactory results were obtained.     

Ionic‐liquid‐based dispersive liquid–liquid microextraction combined with magnetic solid‐phase extraction for the determination of aflatoxins B1, B2, G1, and G2 in animal feeds by high‐performance liquid chromatography with fluorescence detection

A novel two‐step extraction technique combining ionic‐liquid‐based dispersive liquid–liquid microextraction with magnetic solid‐phase extraction was developed for the preconcentration and separation of aflatoxins in animal feedstuffs before high‐performance liquid chromatography coupled with fluorescence detection. In this work, ionic liquid 1‐octyl‐3‐methylimidazolium hexafluorophosphate was used as the extractant in dispersive liquid–liquid microextraction, and hydrophobic pelargonic acid modified Fe₃O₄ magnetic nanoparticles as an efficient adsorbent were applied to retrieve the aflatoxins‐containing ionic liquid. Notably, the target of magnetic nanoparticles was the ionic liquid rather than the aflatoxins. Because of the rapid mass transfer associated with the dispersive liquid–liquid microextraction and magnetic solid phase steps, fast extraction could be achieved. The main parameters affecting the extraction recoveries of aflatoxins were investigated and optimized. Under the optimum conditions, vortexing at 2500 rpm for 1 min in the dispersive liquid–liquid microextraction and magnetic solid‐phase extraction and then desorption by sonication for 2 min with acetonitrile as eluent. The recoveries were 90.3–103.7% with relative standard deviations of 3.2–6.4%. Good linearity was observed with correlation coefficients ranged from 0.9986 to 0.9995. The detection limits were 0.632, 0.087, 0.422 and 0.146 ng/mL for aflatoxins B₁, B2, G1, and G2, respectively. The results were also compared with the pretreatment method carried out by conventional immunoaffinity columns.     

Centrifugeless ultrasound‐assisted emulsification microextraction based on salting‐out phenomenon followed by high‐performance liquid chromatography for the simple determination of phthalate esters in aqueous samples

A fast, sensitive, and centrifugeless ultrasound‐assisted emulsification microextraction followed by a high‐performance liquid chromatography method is developed for the determination of some phthalate esters in aqueous samples. In this method, a simple approach is followed to eliminate the centrifugation step in dispersive liquid–liquid microextraction using an organic solvent whose melting point is near the ambient temperature, consumption of the extracting solvent is efficiently reduced, and the overall extraction time was found to be only 7 min. The variables affecting the method are optimized. Under the optimal experimental conditions (75 μL of 1‐undecanol, a flow rate of 2.0 mL/min, and an ultrasound irradiation of 1 min), the proposed method exhibits good preconcentration factors (52–97), low limits of detection (1.0–5.0 ng/mL), and linearities in the range of 5–1500 ng/mL (r ² ≥ 0.995). Finally, the method is successfully applied to the analysis of phthalate esters in the drinking and river water samples. To study the probable release of the phthalate esters from a polyethylene container into boiling water, the boiling water exposed to the polyethylene container was analyzed by the proposed method.     

Simple determination of some antidementia drugs in wastewater and human plasma samples by tandem dispersive liquid–liquid microextraction followed by high‐performance liquid chromatography

In this work, a simple method, namely, tandem dispersive liquid–liquid microextraction, with a high sample clean‐up is applied for the rapid determination of the antidementia drugs rivastigmine and donepezil in wastewater and human plasma samples. This method, which is based on two consecutive dispersive microextractions, is performed in 7 min. In the method, using a fast back‐extraction step, the applicability of the dispersive microextraction methods in complicated matrixes is conveniently improved. This step can be performed in less than 2 min, and very simple tools are required for this purpose. To achieve the best extraction efficiency, optimization of the variables affecting the method was carried out. Under the optimized experimental conditions, the relative standard deviations for the method were in the range of 6.9–8.7%. The calibration curves were obtained in the range of 2–1100 ng/mL with good correlation coefficients, higher than 0.995, and the limits of detection ranged between 0.5 and 1.0 ng/mL.     

Magnetic core micelles as a nanosorbent for the efficient removal and recovery of three organophosphorus pesticides from fruit juice and environmental water samples

Sodium dodecyl sulfate coated amino‐functionalized magnetic iron oxide nanoparticles were used as an efficient adsorbent for rapid removal and preconcentration of three important organophosphorus pesticides, chlorpyrifos, diazinon and phosalone, by ultrasound‐assisted dispersive magnetic solid‐phase microextraction. Fabrication of amino‐functionalized magnetic nanoparticles was certified by characteristic analyses, including Fourier transform infrared spectroscopy, thermogravimetric analysis, scanning electron microscopy, and transmission electron microscopy. Affecting parameters on the removal efficiency were investigated and optimized through half‐fractional factorial design and Doehlert design, respectively. The analysis of analytes was performed by high‐performance liquid chromatography with ultraviolet detection. Under the optimum conditions, extraction recoveries for 20 ng/mL of organophosphorus pesticides were in the range of 84–97% with preconcentration factors in the range of 134–155. Replicating the experiment in above condition for five times gave the relative standard deviations <6%. The calibration curves showed high linearity in the range of 0.2–700 ng/mL and the limits of detection were in the range of 0.08–0.13 ng/mL. The proposed method was successfully applied for both removal and trace determination of these three organophosphorus pesticides in environmental water and fruit juice samples.     

Cyclodextrin‐assisted dispersive liquid–liquid microextraction for the preconcentration of carbamazepine and clobazam with subsequent sweeping micellar electrokinetic chromatography

A new version of dispersive liquid–liquid microextraction, namely, cyclodextrin‐assisted dispersive liquid–liquid microextraction, with subsequent sweeping micellar electrokinetic chromatography has been developed for the preconcentration and sensitive detection of carbamazepine and clobazam. α‐Cyclodextrin and chloroform were used as the dispersive agent and extraction solvent, respectively. After the extraction, carbamazepine and clobazam were analyzed using micellar electrokinetic chromatography with ultraviolet detection. The detection sensitivity was further enhanced using the sweeping technique. Under optimal extraction and stacking conditions, the calibration curves of carbamazepine and clobazam were linear over a concentration range of 2.0–200.0 ng/mL. The method detection limits at a signal‐to‐noise ratio of 3 were 0.6 and 0.5 ng/mL with sensitivity enhancement factors of 3575 and 4675 for carbamazepine and clobazam, respectively. This developed method demonstrated high sensitivity enhancement factors and was successfully applied to the determination of carbamazepine and clobazam in human urine samples. The precision and accuracy for urine samples were less than 4.2 and 6.9%, respectively.     

Determination of four sulfonylurea herbicides in tea by matrix solid‐phase dispersion cleanup followed by dispersive liquid–liquid microextraction

Matrix solid‐phase dispersion combined with dispersive liquid–liquid microextraction has been developed as a new sample pretreatment method for the determination of four sulfonylurea herbicides (chlorsulfuron, bensulfuron‐methyl, chlorimuron‐ethyl, and pyrazosulfuron) in tea by high‐performance liquid chromatography with diode array detection. The extraction and cleanup by matrix solid‐phase dispersion was carried out by using CN‐silica as dispersant and carbon nanotubes as cleanup sorbent eluted with acidified dichloromethane. The eluent of matrix solid‐phase dispersion was evaporated and redissolved in 0.5 mL methanol, and used as the dispersive solvent of the following dispersive liquid–liquid microextraction procedure for further purification and enrichment of the target analytes before high‐performance liquid chromatography analysis. Under the optimum conditions, the method yielded a linear calibration curve in the concentration range from 5.0 to 10 000 ng/g for target analytes with a correlation coefficients (r²) ranging from 0.9959 to 0.9998. The limits of detection for the analytes were in the range of 1.31–2.81 ng/g. Recoveries of the four sulfonylurea herbicides at two fortification levels were between 72.8 and 110.6% with relative standard deviations lower than 6.95%. The method was successfully applied to the analysis of four sulfonylurea herbicides in several tea samples.     

Optimization of dispersive liquid–liquid microextraction based on the solidification of floating organic droplets using an orthogonal array design and its application for the determination of fungicide concentrations in environmental water samples

A dispersive liquid–liquid microextraction method based on the solidification of floating organic droplets was developed as a simple and sensitive method for the simultaneous determination of the concentrations of multiple fungicides (triazolone, chlorothalonil, cyprodinil, and trifloxystrobin) in water by high‐performance liquid chromatography with variable‐wavelength detection. After an approach varying one factor at a time was used, an orthogonal array design [L₂₅ (5⁵)] was employed to optimize the method and to determine the interactions between the parameters. The significance of the effects of the different factors was determined using analysis of variance. The results indicated that the extraction solvent volume significantly affects the efficiency of the extraction. Under optimal conditions, the relative standard deviation (n = 5) varied from 2.3 to 5.5% at 0.1 μg/mL for each analyte. Low limits of detection were obtained and ranged from 0.02 to 0.2 ng/mL. In addition, the proposed method was applied to the analysis of fungicides in real water samples. The results show that the dispersive liquid–liquid microextraction based on the solidification of floating organic droplets is a potential method for detecting fungicides in environmental water samples, with recoveries of the target analytes ranging from 70.1 to 102.5%.     

Application of tandem dispersive liquid–liquid microextraction for the determination of doxepin,citalopram, and fluvoxamine in complicated samples

A new type of dispersive liquid–liquid microextraction is used for the determination of doxepin, citalopram, and fluvoxamine in aqueous matrices. This method is based upon the tandem utilization of dispersive liquid–liquid microextraction, and by providing a high sample clean‐up, it efficiently improves the applicability of the method in complicated matrices. For this purpose, in the first step, the analytes contained in an aqueous sample solution (8.0 mL) were extracted into an organic solvent, and then these analytes were simply back‐extracted into an aqueous acceptor phase (50 μL). The overall extraction time was 7 min, and very simple tools were required for this aim. Optimization of the variables affecting the method such as the type and volume of the organic solvent used and effect of ionic strength was carried out to achieve the best extraction efficiency. Under the optimized experimental conditions, tandem dispersive liquid–liquid microextraction with high‐performance liquid chromatography and UV detection showed a good linearity in the range of 10–5000 ng/mL. The limits of detection were in the range of 3–10 ng/mL. The Intra‐day precisions (relative standard deviation) were 9.2, 4.5, and 4.8, and the recoveries were 58.5, 52.9, and 39.3% for citalopram, doxepin, and fluvoxamine, respectively.     

Preconcentration and determination of 2‐mercaptobenzimidazole by dispersive liquid–liquid microextraction and experimental design

A method was developed to determine 2‐mercaptobenzimidazole in water and urine samples using dispersive liquid–liquid microextraction technique coupled with ultraviolet–visible spectrophotometry. It was essential to peruse the effect of all parameters that can likely influence the performance of extraction. The influence of parameters, such as dispersive and extraction solvent volume and sample volume, on dispersive liquid–liquid microextraction was studied. The optimization was carried out by the central composite design method. The central composite design optimization method resulted in 1.10 mL dispersive solvent, 138.46 μL extraction solvent, and 4.46 mL sample volume. Under the optimal terms, the calibration curve was linear over the range of 0.003–0.18 and 0.007–0.18 μg/mL in water and urine samples, respectively. The limit of detection and quantification of the proposed approach for 2‐mercaptobenzimidazole were 0.013 and 0.044 μg/mL in water samples and 0.016 and 0.052 μg/mL in urine samples, respectively. The method was successfully applied to determination of 2‐mercaptobenzimidazole in urine and water samples.     

Development of a simple and valid method for the trace determination of phthalate esters in human plasma using dispersive liquid–liquid microextraction coupled with gas chromatography–mass spectrometry

A new method was developed for the trace determination of phthalic acid esters in plasma using dispersive liquid–liquid microextraction and gas chromatography with mass spectrometry analysis. Plasma proteins were efficiently precipitated by trichloroacetic acid and then a mixture of chlorobenzene (as extraction solvent) and acetonitrile (as dispersive solvent) rapidly injected to clear supernatant using a syringe. After centrifuging, chlorobenzene sedimented at the bottom of the test tube. 1 μL of this sedimented phase was injected into the gas chromatograph for phthalic acid esters analysis. Different factors affecting the extraction performance, such as the type of extraction and dispersive solvent, their volume, extraction time, and the effects of salt addition were investigated and optimized. Under the optimum conditions, the enrichment factors and extraction recoveries were satisfactory and ranged between 820–1020 and 91–97%, respectively. The linear range was wide (50–1000 ng/mL) and limit of detection was very low (1.5–2.5 ng/mL for all analytes). The relative standard deviations for analysis of 1 μg/mL of the analytes were between 3.2–6.1%. Salt addition showed no significant effect on extraction recovery. Finally, the proposed method was successfully utilized for the extraction and determination of the phthalic acid esters in human plasma samples and satisfactory results were obtained.     

New approach applying a pet fish air pump in liquid‐phase microextraction for the determination of Sudan dyes in food samples by HPLC

A new approach applying a pet fish air pump is introduced to develop an extraction method, namely, air‐pump‐enhanced emulsion, followed by salt‐assisted emulsion breaking based on solidified floating organic drop microextraction for the extraction and preconcentration of Sudan I–IV before high‐performance liquid chromatography. The applicability of this method was successfully demonstrated by determination of these dyes in four chili products that include chili powder, chili oil, chili sauce, and chili paste. An enrichment factor of 62 was obtained only with a sample solution of 5 mL. A linear range of 0.5–2500 ng/mL was obtained with a limit of detection of 0.16–0.24 ng/mL and recovery of 90–110%. This method is superior to other liquid–liquid extraction methods, as is simple, rapid, environmental friendly, and its phase separation needs no centrifugation. It also needs no disperser solvent and requires less organic solvent, and satisfies the criteria to be called as a green extraction. Therefore, this facile extraction method can be successfully applied in the determination of Sudan dyes in food samples.     

Dispersive solid‐phase extraction followed by vortex‐assisted dispersive liquid–liquid microextraction based on the solidification of a floating organic droplet for the determination of benzoylurea insecticides in soil and sewage sludge

A novel dispersive solid‐phase extraction combined with vortex‐assisted dispersive liquid–liquid microextraction based on solidification of floating organic droplet was developed for the determination of eight benzoylurea insecticides in soil and sewage sludge samples before high‐performance liquid chromatography with ultraviolet detection. The analytes were first extracted from the soil and sludge samples into acetone under optimized pretreatment conditions. Clean‐up of the extract was conducted by dispersive solid‐phase extraction using activated carbon as the sorbent. The vortex‐assisted dispersive liquid–liquid microextraction based on solidification of floating organic droplet procedure was performed by using 1‐undecanol with lower density than water as the extraction solvent, and the acetone contained in the solution also acted as dispersive solvent. Under the optimum conditions, the linearity of the method was in the range 2–500 ng/g with correlation coefficients (r) of 0.9993–0.9999. The limits of detection were in the range of 0.08–0.56 ng/g. The relative standard deviations varied from 2.16 to 6.26% (n = 5). The enrichment factors ranged from 104 to 118. The extraction recoveries ranged from 81.05 to 97.82% for all of the analytes. The good performance has demonstrated that the proposed methodology has a strong potential for application in the multiresidue analysis of complex matrices.